UDC 517.947.42

ON THE SMOOTHNESS OF THERMAL POTENTIALS. III*)

L. I. KAMYNIN

§ 14. PROOF OF THEOREM 18

By the maximum principle (cf. § 5 [1]), from (10.6), (10.43), and Theorem 8 of § 2 [1] it follows that it is sufficient to prove the validity of the relations

\[ \left|\overline{P}_{11}(x_1,t+\Delta t)-\overline{P}_{11}(x_1,t)\right| \leq (C)|\varphi|_{\alpha}|\Delta t|^{\frac{1+\alpha'}{2}}, \]

\[ \left|\overline{P}_{11}(x,t)\right| \leq (C)|\varphi|_{\alpha} t^{\frac{1+\alpha}{2}}, \tag{14.1} \]

\[ \overline{\frac{\partial P_{11}(x,t)}{\partial x_i}} \in H^{0,\alpha',\alpha/2}(\Gamma), \qquad \left|\overline{\frac{\partial P_{11}(x,t)}{\partial x_i}}\right| \leq (C)|\varphi|_{\alpha} t^{\alpha/2}, \tag{14.2} \]

where (see (10.7), Remark 10 of § 10, and Remark 12 of § 12)

\[ \overline{P}_{11}(x,t)\equiv \overline{P}_{11}(x_1,t)= \]

\[ =\int_{t_1}^{t} d\tau \int_{-\infty}^{+\infty} \varphi(\xi_1,\tau)\,\overline{p}\,(r(\xi,x),t-\tau)\,d\xi_1 . \tag{14.3} \]

Put (cf. (13.5), (13.16), (13.17))

\[ \overline{P}_{11}(x_1,t)= \sum_{i=1}^{2} \left( \overline{P}_{11}^{(0i)}(x_1,t)+ \overline{P}_{11}^{(1i)}(x_1,t) \right), \]

\[ \overline{P}_{11}^{(1i)}(x_1,t) \equiv \overline{P}_{11}^{(i)}(x_1,t)- \overline{P}_{11}^{(0i)}(x_1,t), \]

where (see (13.4), (13.12), (13.22))

\[ \overline{P}_{11}^{(01)}(x_1,t)= \int_{t_1}^{t}\varphi_1^*(x_1,\tau)\,d\tau \int_{-\infty}^{+\infty} \xi_1 G_{0,3/2}(\xi_1,t-\tau)\,d\xi_1 = \]

\[ = -4\int_{t_1}^{t} (t-\tau)^{-1/2}\varphi_1(x_1,\tau)\nu_2^{-2}(x_1,\tau)\Phi_0(\tau)\,d\tau, \]

\[ \overline{P}_{11}^{(02)}(x_1,t)= -\int_{t_1}^{t}\varphi_{20}^*(x_1,\tau)\,d\tau \int_{-\infty}^{+\infty} G_{0,3/2}(\xi_1,t-\tau)\,d\xi_1 = \]

*) For the beginning of the article, see Differential Equations, 2, No. 10, 1966.

\[ = \pi \int_{t_1}^{t} (t-\tau)^{-1} \varphi_{20}^{*}(x_1,\tau)\,d\tau, \]

\[ \overline P_{11}^{(1)}(x_1,t) = \int_{t_1}^{t} d\tau \int_{-\infty}^{+\infty} \varphi_1^{*}(\xi_1,\tau)\,\xi_1\,G_{3/2}(r(\xi,x),t-\tau)\,d\xi_1, \]

\[ \overline P_{11}^{(2)}(x_1,t) = -\int_{t_1}^{t} d\tau \int_{-\infty}^{+\infty} \varphi_{20}^{*}(\xi_1,\tau)\,G_{3/2}(r(\xi,x),t-\tau)\,d\xi_1, \]

where

\[ \varphi_i^{*}(\xi_1,\tau)=\varphi_0^{*}(\xi_1,\tau)\mu_i(\xi_1,\tau) \qquad (i=1,2), \]

\[ \varphi_0^{*}(\xi_1,\tau)=\varphi(\xi_1,\tau)\bigl(\mu(\xi_1,\tau),\,N(\xi_1,\tau)\bigr), \]

\[ \varphi_{20}^{*}(\xi_1,\tau)=\varphi_2^{*}(\xi_1,\tau)\bigl(\psi(x_1,t)-\psi(\xi_1,\tau)\bigr), \tag{14.4} \]

\[ \varphi^{*}(\xi_1,\tau)=\varphi_0^{*}(\xi_1,\tau)\mu(\xi_1,\tau). \]

By Lemma 3 of § 1 [1],

\[ \left| \overline P_{11}^{(01)}(x_1,t+\Delta t)-\overline P_{11}^{(01)}(x_1,t) \right| \le (C)|\varphi|_\alpha |\Delta t|^{\frac{1+\alpha^0}{2}}. \]

Repeating the arguments of § 13 in deriving (13.26), (13.31), we obtain, by virtue of the conditions of Theorem 18, Lemma 11 of § 10, and Lemma 13 of § 13, the estimates

\[ \left\{ \begin{array}{l} \left| \overline P_{11}^{(02)}(x_1,t+\Delta t)-\overline P_{11}^{(02)}(x_1,t) \right|,\\[4pt] \left| \overline P_{11}^{(1i)}(x_1,t+\Delta t)-\overline P_{11}^{(1i)}(x_1,t) \right| \end{array} \right\} \le (C)|\varphi|_{\alpha'}|\Delta t|^{\frac{1+\alpha'}{2}} \qquad (i=1,2) \]

\[ \left| \overline P_{11}^{(ij)}(x_1,t) \right| \le (C)|\varphi|_{\alpha}\,t^{\frac{1+\alpha}{2}} \qquad (i=0,1;\ j=1,2), \]

whence (14.1) follows.

Proceeding to the derivation of (14.2), we note first of all that from Lemma 10 of § 10 (see also Remark 10 of § 10 and (10.14)—(10.16)) there follows the equality

\[ \frac{\partial p(r(y,\overline x),t-\tau)}{\partial \overline x_2} = -\nu_1(y,\tau)\nu_2^{-1}(y,\tau) \frac{\partial p(r(y,\overline x),t-\tau)}{\partial \overline x_1} + \]

\[ +\nu_2^{-1}(y,\tau) \sum_{j=1}^{2} \bigl[ N_j(y,\tau)-(\nu(y,\tau),N(y,\tau))\nu_j(y,\tau) \bigr] \times \]

\[ \times \frac{\partial}{\partial \overline x_j} g_{0,\frac n2}(r(y,\overline x),t-\tau), \tag{14.5} \]

therefore, by virtue of § 5 [1], it suffices to consider the function

\[ P_0(\overline x,t)\equiv -\frac{\partial P_{11}(\overline x,t)}{\partial x_1} = \int_{t_1}^{t} d\tau \int_{-\infty}^{+\infty} \varphi(\xi_1,\tau) \frac{\partial p(r(\xi,\overline x),t-\tau)}{\partial \overline x_1} \,d\xi_1. \tag{14.6} \]

With the aid of the vectors \(\nu(x_1,\tau)\), \(\mu(x_1,\tau)\), \(\nu(\xi_1,\tau)\), \(\mu(\xi_1,\tau)\), construct, for \(0\le z\le 1\), a family of vectors

\[ v(z)=\{v_1(z),\,v_2(z)\},\qquad \mu(z)=\{-v_2(z),\,v_1(z)\}, \]

where

\[ v_1(z)=\cos\bigl(\gamma_0+z(\gamma-\gamma_0)\bigr),\qquad v_2(z)=\sin\bigl(\gamma_0+z(\gamma-\gamma_0)\bigr), \]

\[ \cos\gamma_0=v_1(x_1,\tau),\qquad \sin\gamma_0=v_2(x_1,\tau), \tag{14.7} \]

\[ \cos\gamma=v_1(\xi_1,\tau),\qquad \sin\gamma=v_2(\xi_1,\tau). \]

Obviously,

\[ v(0)=v(x_1,\tau),\qquad v(1)=v(\xi_1,\tau), \]

\[ (v(z),v(z))=(\mu(z),\mu(z))=1,\qquad (v(z),\mu(z))=0, \tag{14.8} \]

\[ r^2(\xi,\bar x)=(\mu(z),r_{\xi x})^2+(v(z),r_{\xi x})^2, \]

\[ \frac{\partial v(z)}{\partial z}=(\gamma-\gamma_0)\mu(z), \]

\[ \frac{\partial \mu(z)}{\partial z}=-(\gamma-\gamma_0)v(z). \tag{14.9} \]

And, by virtue of the condition \(v_j\in H^{0,\beta,\beta/2}(\Gamma)\) (10.13), (13.2), it follows from (14.7) that

\[ |\gamma-\gamma_0|\leq (C)\rho^\beta . \tag{14.10} \]

Denote by \(G_m^{(z)}(r(\xi,\bar x),\,t-\tau)\) the function obtained from (10.15) by replacing \(v(\xi_1,\tau)\), \(\mu(\xi_1,\tau)\) by \(v(z)\), \(\mu(z)\), respectively. We note that, by virtue of (14.9),

\[ \frac{\partial G_m^{(z)}(r(\xi,\bar x),\,t-\tau)}{\partial z} = \frac12(\gamma-\gamma_0)(\mu(z),r_{\xi x})\times \]

\[ {}\times\Bigl[(v(z),r_{\xi x})G_{m+1}^{(z)}(r(\xi,\bar x),\,t-\tau) + g_{0,m+\frac12}(r(\xi,\bar x),\,t-\tau)\Bigr] \tag{14.11} \]

and, by the mean-value theorem,

\[ G_m^{(1)}(r(\xi,\bar x),\,t-\tau)-G_m^{(0)}(r(\xi,\bar x),\,t-\tau) = \]

\[ =\int_0^1 \frac{\partial}{\partial z}G_m^{(z)}(r(\xi,\bar x),\,t-\tau)\,dz. \tag{14.12} \]

Therefore, by Lemma 4 § 1 [1], Lemma 11 § 10, from (14.10)—(14.12) we have the estimate

\[ \bigl|G_m^{(1)}(r(\xi,\bar x),\,t-\tau)-G_m^{(0)}(r(\xi,\bar x),\,t-\tau)\bigr|\leq \]

\[ \leq (C)g_{0,m-\beta/2}(\chi r(\xi,\bar x),\,2(t-\tau)). \tag{14.13} \]

Let (12.15) hold. Then

\[ r_{\xi x}=\{x_1-\xi_1,\ \delta+\psi(x_1,t)-\psi(\xi_1,\tau)\}, \]

\[ \frac{\partial r_{\xi x}}{\partial x_1} = \left\{1,\frac{\partial\psi(x_1,t)}{\partial x_1}\right\}, \qquad \frac{\partial r_{\xi x}}{\partial \xi_1} = \left\{-1,-\frac{\partial\psi(\xi_1,\tau)}{\partial \xi_1}\right\}, \tag{14.14} \]

\[ \frac{\partial r_{\bar{\xi}x}}{\partial x_1} = -\frac{\partial r_{\bar{\xi}x}}{\partial \xi_1} + \left\{0,\, \frac{\partial \psi(x_1,t)}{\partial x_1} - \frac{\partial \psi(\xi_1,\tau)}{\partial \xi_1} \right\}. \tag{14.14} \]

Set (see (14.4))

\[ \varphi(\xi_1,\tau)\,\frac{\partial}{\partial x_1}\, p\bigl(r(\xi,\bar{x}),t-\tau\bigr) = \bigl(\varphi^*(x_1,\tau),\,q_0^{(3/2)}(r(\xi,\bar{x}),t-\tau)\bigr) + \]

\[ + \bigl(\varphi^*(x_1,\tau)-\varphi^*(\xi_1,\tau),\, q_1^{(3/2)}(r(\xi,\bar{x}),t-\tau)\bigr) - \]

\[ -\bigl(\varphi^*(x_1,\tau),\, q_{10}^{(3/2)}(r(\xi,\bar{x}),t-\tau)\bigr), \tag{14.15} \]

where the notations have been introduced \((\bar{x}_1\equiv x_1)\)

\[ q_1^{(m)}(r(\xi,\bar{x}),t-\tau) \equiv \frac{\partial}{\partial x_1} \bigl[r_{\bar{\xi}x}G_m^{(1)}(r(\xi,\bar{x}),t-\tau)\bigr], \]

\[ q_0^{(m)}(r(\xi,\bar{x}),t-\tau) \equiv \frac{\partial}{\partial \xi_1} \bigl[r_{\bar{\xi}x}G_m^{(0)}(r(\xi,\bar{x}),t-\tau)\bigr], \tag{14.16} \]

\[ q_{10}^{(m)}(r(\xi,\bar{x}),t-\tau) = \sum_{i=0}^{1} q_i^{(m)}(r(\xi,\bar{x}),t-\tau). \tag{14.17} \]

From (10.42) (where \(x=\bar{x}\)) and (see (14.8))

\[ \frac{\partial}{\partial \xi_1} G_m^{(0)}(r(\xi,\bar{x}),t-\tau) = -\frac{1}{2}(\mu(0),r_{\bar{\xi}x})\times \]

\[ \times \left(\mu(0),-\frac{\partial r_{\bar{\xi}x}}{\partial \xi_1}\right) G_{m+1}^{(0)}(r(\xi,\bar{x}),t-\tau) + \]

\[ +\frac{1}{2} \left(\nu(0),-\frac{\partial r_{\bar{\xi}x}}{\partial \xi_1}\right) g_{0,m+\frac12}(r(\xi,\bar{x}),t-\tau), \tag{14.18} \]

by virtue of (14.14) we have

\[ q_{10}^{(m)}(r(\xi,\bar{x}),t-\tau) = \{q_{101}^{(m)}(\bar{x},t),\,q_{102}^{(m)}(\bar{x},t)\} - \sum_{i=3}^{5}q_{10i}^{(m)}(\bar{x},t), \]

\[ q_{101}^{(m)}(\bar{x},t) \equiv G_m^{(1)}(r(\xi,\bar{x}),t-\tau) - G_m^{(0)}(r(\xi,\bar{x}),t-\tau), \tag{14.19} \]

\[ q_{102}^{(m)}(\bar{x},t) \equiv \frac{\partial\psi(x_1,t)}{\partial x_1} G_m^{(1)}(r(\xi,\bar{x}),t-\tau) - \]

\[ - \frac{\partial\psi(\xi_1,\tau)}{\partial \xi_1} G_m^{(0)}(r(\xi,\bar{x}),t-\tau), \]

\[ q_{103}^{(m)}(\bar{x},t) \equiv \frac{1}{2}r_{\bar{\xi}x} \left( \frac{\partial\psi(x_1,t)}{\partial x_1} - \frac{\partial\psi(\xi_1,\tau)}{\partial \xi_1} \right) \times \]

\[ \times \bigl[(\mu(x_1,\tau),r_{\bar{\xi}x})\mu_2(x_1,\tau) G_{m+1}^{(0)}(r(\xi,\bar{x}),t-\tau) + \]

\[ +\,v_2(x_1,\tau)g_{0,m+\frac12}(r(\xi,\bar x),t-\tau)], \]

\[ \begin{aligned} q_{104}^{(m)}(\bar x,t) &=\frac12\,r_{\xi x}^{-}\Biggl\{\Bigl[\bigl(\mu(\xi_1,\tau)-\mu(x_1,\tau),r_{\xi x}^{-}\bigr) \Bigl(\mu(\xi_1,\tau),\frac{\partial r_{\xi x}^{-}}{\partial x_1}\Bigr) \\ &\quad+\bigl(\mu(x_1,\tau),r_{\xi x}^{-}\bigr) \Bigl(\mu(\xi_1,\tau)-\mu(x_1,\tau),\frac{\partial r_{\xi x}^{-}}{\partial x_1}\Bigr)\Bigr] G_{m+1}^{(1)}(r(\xi,\bar x),t-\tau) \\ &\quad+\Bigl(v(\xi_1,\tau)-v(x_1,\tau),\frac{\partial r_{\xi x}^{-}}{\partial x_1}\Bigr) g_{0,m+\frac12}(r(\xi,\bar x),t-\tau)\Biggr\}, \end{aligned} \]

\[ q_{105}^{(m)}(\bar x,t) = \frac12\,r_{\xi x}^{-}\bigl(\mu(x_1,\tau),r_{\xi x}^{-}\bigr) \Bigl(\mu(x_1,\tau),-\frac{\partial r_{\xi x}^{-}}{\partial x_1}\Bigr) q_{101}^{(m+1)}(\bar x,t). \]

Lemma 14. If \((\Gamma)\) is of type \(\Pi_{1,\alpha,\alpha/2}^{0,1,\frac{1+\alpha}{2}}\) and \(v_j\in H^{0,\beta,\beta/2}(\Gamma)\) \((0<\alpha\le \beta\le 1)\), then, under (12.15), (14.14), (14.16), (14.17),

\[ \left|q_1^{(m)}(r(\xi,\bar x),t-\tau)\right| \le (C)g_{0,m}(\chi r(\xi,\bar x),2(t-\tau)), \tag{14.20} \]

\[ \left|q_{10}^{(m)}(r(\xi,\bar x),t-\tau)\right| \le (C)g_{0,m-\beta/2}(\chi r(\xi,\bar x),2(t-\tau)), \tag{14.21} \]

\[ \begin{aligned} &\left|q_1^{(m)}(r(\xi,\bar x),t-\tau) -\bar q_1^{(m)}(r(\xi,x),t-\tau)\right| \le \\ &\qquad\le (C)\delta\,g_{0,m+1/2}(\chi\rho,2(t-\tau)) \int_0^1 g_{0,0}(\chi|\theta\delta-\psi(\xi_1,\tau)|,2(t-\tau))\,d\theta, \end{aligned} \tag{14.22} \]

\[ \begin{aligned} &\left|q_{10}^{(m)}(r(\xi,\bar x)^*,t-\tau) -\bar q_{10}^{(m)}(r(\xi,x),t-\tau)\right| \le \\ &\qquad\le (C)\delta\,g_{0,m+\frac{1-\alpha}{2}}(\chi\rho,2(t-\tau)) \times \\ &\qquad\quad\times\int_0^1 g_{0,0}(\chi|\theta\delta-\psi(\xi_1,\tau)|,2(t-\tau))\,d\theta. \end{aligned} \tag{14.23} \]

Proof. Estimates (14.20), (14.21) follow from (10.42), (14.19), (14.13) with the aid of Lemma 4 § 1 [1] and Lemma 11 § 10. From (12.17), (10.42), and (see (14.16)) the representation following from the mean-value theorem is

\[ q_1^{(m)}(r(\xi,\bar x),t-\tau) -\bar q_1^{(m)}(r(\xi,x),t-\tau) = \]

\[ = \int_0^1 \frac{\partial}{\partial\theta} \left[ r_{\xi x(\theta)}\,q_1^{(m)}(r(\xi,\bar x(\theta)),t-\tau) \right]\,d\theta \]

With the aid of Lemma 4 § 1 [1] and Lemma 11 § 10 we obtain (14.22). To derive (14.23) we use the representation, by the mean-value theorem, of
\(q_{10i}^{(m)}(\bar{x}, t)-\bar q_{10i}^{(m)}(x,t)\) from (14.19) (for example, from (14.12) it follows that

\[ q_{101}^{(m)}(\bar{x}, t)-\bar q_{101}^{(m)}(x,t)= \]

\[ =\int_0^1 \frac{\partial}{\partial \theta} \left[ \int_0^1 \frac{\partial}{\partial z} G_m^{(z)}\bigl(r(\xi,\bar{x}(\theta)),\,t-\tau\bigr)\,dz \right]d\theta). \]

The use of (14.9), (14.10), (1.14), (1.17) (for \(\varphi\) and with \(\alpha=\beta\) for \(v_j\)) § 1 [1] gives, with the aid of Lemma 4 § 1 [1] and (see (14.8)) Lemma 11 § 10, the estimate (14.23).

With the aid of integration by parts from (14.6), (12.5), (14.15), arguing as in the derivation of (5.27) § 5 [1], we obtain the representation (see (14.16), (14.17))

\[ P_0(\bar{x},t)=\sum_{i=1}^2 P_{0i}(\bar{x},t), \tag{14.24} \]

where

\[ P_{01}(\bar{x},t)=\int_{t_1}^{t}d\tau\int_{-\infty}^{+\infty} \bigl(\varphi^*(x_1,\tau)-\varphi^*(\xi_1,\tau),\, q_1^{(3/2)}(r(\xi,\bar{x}),t-\tau)\bigr)\,d\xi_1, \]

\[ \tag{14.25} \]

\[ P_{02}(\bar{x},t)= -\int_{t_1}^{t}d\tau\int_{-\infty}^{+\infty} \bigl(\varphi^*(x_1,\tau),\, q_{10}^{(3/2)}(r(\xi,\bar{x}),t-\tau)\bigr)\,d\xi_1. \]

From (14.25), with the aid of the estimates (14.20), (14.21) of Lemma 14 and the condition
\(\varphi^*\in H^{0,\alpha,\alpha/2}(\Gamma)\), we have

\[ |P_{0i}(\bar{x},t)|\le (C)|\varphi|_{\alpha}\,t^{\alpha/2} \qquad (i=1,2). \tag{14.26} \]

Moreover, the estimates (14.22), (14.23) of Lemma 14 make it possible, by the method of § 5 [1] applied in deriving (5.30) (see also the derivation of estimate (12.20)), to obtain the inequality (see (12.15))

\[ |P_{0i}(\bar{x},t)-\bar P_{0i}(x,t)|\le (C)|\varphi|_{\alpha}\delta^{\alpha} \qquad (i=1,2). \tag{14.27} \]

Lemma 15. If the conditions of Lemma 14 are satisfied for \(\Gamma\) and \(v\), then under (10.31)

\[ \left| \bar q_1^{(m)}(r(\xi,\hat{x}),t+\Delta t-\tau) - \bar q_1^{(m)}(r(\xi,x),t-\tau) \right| \le \]

\[ \le (C)\left[ (\Delta t)^{\alpha/2}(t-\tau)^{-m} + (\Delta t)^{\frac{1+\alpha}{2}}(t-\tau)^{-m-1/2} + \right. \tag{14.28} \]

\[ \left. +\Delta t\,(t-\tau)^{-m-1} \right]g_{0,0}\bigl(x\rho,\,2(t+\Delta t-\tau)\bigr), \]

\[ \left| \bar q_{10}^{(m)}(r(\xi,\hat{x}),t+\Delta t-\tau) - \bar q_{10}^{(m)}(r(\xi,x),t-\tau) \right| \le \tag{14.29} \]

\[ \le (C)\left[ (\Delta t)^{\alpha/2}(t-\tau)^{-m+\beta/2} + (\Delta t)^{\frac{1+\alpha}{2}}(t-\tau)^{-m+\frac{\beta-1}{2}} + \right. \]

\[ +\Delta t(t-\tau)^{-m-1+\frac{\beta}{2}} g_{\nu,0}\bigl(\chi\rho,\ 2(t+\Delta t-\tau)\bigr). \tag{14.29} \]

The proof of Lemma 15 is carried out according to the scheme of deriving (10.35), (10.36), using the representations by the mean-value theorem for the differences
\(\bar q_{10i}^{(m)}(\hat x,t+\Delta t)-\bar q_{10i}^{(m)}(x,t)\) from (14.19), if one introduces the vectors (10.40) and

\[ \frac{\partial r_{\hat \xi x(\theta)}}{\partial \theta} = \{0,\ \psi(x_1,t+\Delta t)-\psi(x_1,t)\}, \]

\[ \frac{\partial r_{\hat \xi x(\theta)}}{\partial x_1} = \left\{ 1,\ \frac{\partial\psi(x_1,t)}{\partial x_1} + \theta\left( \frac{\partial\psi(x_1,t+\Delta t)}{\partial x_1} - \frac{\partial\psi(x_1,t)}{\partial x_1} \right) \right\} \tag{14.30} \]

and takes into account (14.10), (14.11), Lemma 4 § 1 [1], Lemma 11 § 10 (see (14.8)) and the estimates (1.12), (1.13) § 1 [1].

From (14.25), with the aid of Lemma 15, the estimates (14.26), and

\[ |\varphi^*(x_1,\tau)-\varphi^*(\xi_1,\tau)| \leq (C)|\varphi|_\alpha \rho^\alpha \tag{14.31} \]

we obtain by the usual device (cf. the derivation of (13.48)) the estimate

\[ P_{0i}(\hat x,t+\Delta t)-\bar P_{0i}(x,t) \leq (C)|\varphi|_\alpha |\Delta t|^{\alpha/2} \qquad (i=1,2). \tag{14.32} \]

Lemma 16. If the hypotheses of Lemma 14 are satisfied for \(\Gamma\) and \(\nu\), then under (10.37), (12.4), (12.11) the estimates (14.20), (14.21) take the form

\[ \left|\bar q_{1}^{(m)}\bigl(r(\xi,x^\delta),t-\tau\bigr)\right| \leq (C)g_{0,m}\bigl(\chi\rho_1,\ 2(t-\tau)\bigr), \tag{14.33} \]

\[ \left|\bar q_{10}^{(m)}\bigl(r(\xi,x^\delta),t-\tau\bigr)\right| \leq (C)g_{0,m-\beta/2}\bigl(\chi\rho_1,\ 2(t-\tau)\bigr) \tag{14.34} \]

and, for \(\rho \geq 2\delta\) (see (12.13)),

\[ \left| \bar q_{1}^{(m)}\bigl(r(\xi,x^\delta),t-\tau\bigr) - \bar q_{1}^{(m)}\bigl(r(\xi,x),t-\tau\bigr) \right| \leq \]

\[ \leq (C)\delta\, g_{0,m+1/2}\left(\frac{\chi\rho}{2},\ 2(t-\tau)\right), \tag{14.35} \]

\[ \left| q_{10}^{(m)}\bigl(r(\xi,x^\delta),t-\tau\bigr) - \bar q_{10}^{(m)}\bigl(r(\xi,x),t-\tau\bigr) \right| \leq \]

\[ \leq (C)\left[ \delta^\beta g_{0,m}\left(\frac{\chi\rho}{2},\ 2(t-\tau)\right) + \delta^{1+\beta} g_{0,m+1/2} \left(\frac{\chi\rho}{2},\ 2(t-\tau)\right) + \delta g_{0,m+\frac{1-\beta}{2}} \left(\frac{\chi\rho}{2},\ 2(t-\tau)\right) \right]. \tag{14.36} \]

Proof. The derivation of the estimates (14.33), (14.34) is similar to the derivation of (14.20), (14.21). For \(\rho \geq 2\delta\), (12.13) holds, which makes it possible to derive (14.35) by the mean-value theorem of Lemma 12 § 10. To derive (14.36), for \(0\leq \theta,\ z\leq 1\) consider the family of vectors (cf. (14.7), (14.10)) \(\nu^\delta(\theta,z)\), \(\mu^\delta(\theta,z)\), where

\[ \nu_1^\delta(\theta,z) = \cos[\gamma_0+z(\gamma-\gamma_0)+\theta(1-z)(\gamma_\delta-\gamma_0)], \]

\[ \mu_1^\delta(\theta,z)=-\nu_2^\delta(\theta,z), \]

\[ v_2^\delta(\theta,z)=\sin[\gamma_0+z(\gamma-\gamma_0)+\theta(1-z)(\gamma_\delta-\gamma_0)], \]

\[ \mu_2^\delta(\theta,z)=v_1^\delta(\theta,z) \]

and, along with (14.7), set

\[ \cos\gamma_\delta=v_1(\delta,\tau),\qquad \sin\gamma_\delta=v_2(\delta,\tau), \tag{14.37} \]

where

\[ \bigl(v^\delta(\theta,z),v^\delta(\theta,z)\bigr) = \bigl(\mu^\delta(\theta,z),\mu^\delta(\theta,z)\bigr)=1, \tag{14.38} \]

\[ \bigl(\mu^\delta(\theta,z),v^\delta(\theta,z)\bigr)=0 \]

and

\[ \frac{\partial v^\delta(\theta,z)}{\partial z} = [\gamma-\gamma_0+\theta(\gamma_0-\gamma_\delta)]\mu^\delta(\theta,z), \]

\[ \frac{\partial \mu^\delta(\theta,z)}{\partial z} = -[\gamma-\gamma_0+\theta(\gamma_0-\gamma_\delta)]v^\delta(\theta,z), \]

\[ \frac{\partial v^\delta(\theta,z)}{\partial\theta} = (1-z)(\gamma_\delta-\gamma_0)\mu^\delta(\theta,z), \]

\[ \frac{\partial}{\partial\theta}\mu^\delta(\theta,z) = -(1-z)(\gamma_\delta-\gamma_0)v^\delta(\theta,z), \]

whence, in view of (14.7), (14.37), we have

\[ \left[\left|\frac{\partial}{\partial\theta}v^\delta(\theta,z)\right|, \left|\frac{\partial\mu^\delta(\theta,z)}{\partial\theta}\right|\right]\leq (C)\delta^\beta, \tag{14.39} \]

\[ \left[\left|\frac{\partial}{\partial z}v^\delta(\theta,z)\right|, \left|\frac{\partial\mu^\delta(\theta,z)}{\partial z}\right|\right]\leq (C)(\rho^\beta+\delta^\beta), \tag{14.40} \]

\[ \left[\left|\frac{\partial^2 v^\delta(\theta,z)}{\partial z\,\partial\theta}\right|, \left|\frac{\partial^2\mu^\delta(\theta,z)}{\partial z\,\partial\theta}\right|\right] \leq (C)\delta^\beta(1+\rho^\beta+\delta^\beta). \tag{14.41} \]

Set (see (10.15))

\[ G_m^{(z,\theta)}\bigl(r(\xi,x^\delta(\theta)),\,t-\tau\bigr) \equiv (t-\tau)^{-m}\times \]

\[ \times \exp\left\{-\frac{\bigl(\mu^\delta(\theta,z),\,r_{\xi x^\delta(\theta)}\bigr)}{4(t-\tau)}\right\} \left\{ \Phi\left( \frac{\bigl(v^\delta(\theta,z),\,r_{\xi x^\delta(\theta)}\bigr)} {2\sqrt{t-\tau}} \right) -\frac{\sqrt{\pi}}{2} \right\}. \]

The use of representations by the mean-value theorem for the differences

\[ \overline{q}_{10i}^{(m)}(x^\delta,t)-\overline{q}_{10i}^{(m)}(x,t) \]

from (14.19) (for example,

\[ \overline{q}_{101}^{(m)}(x^\delta,t)-\overline{q}_{101}^{(m)}(x,t) = \int_0^1 \frac{\partial}{\partial\theta}\times \]

\[ \times \left[ \int_0^1 \frac{\partial}{\partial z} G_m^{(z,\theta)}\bigl(r(\xi,x^\delta(\theta)),t-\tau\bigr)\,dz \right]d\theta \]

) gives, with the aid of (14.38)—(14.41), (12.13), Lemma 4 § 1 [1], and Lemma 11 § 10, the estimate (14.36).

From (14.24), (14.25), with the aid of Lemma 16, the estimates (14.31), and

\[ \left|\varphi^{*}(\delta,\tau)-\varphi^{*}(\xi_1,\tau)\right| \leq (C)|\varphi|_\alpha\left(\rho_1^\alpha+\delta^\alpha\right) \]

by the usual method (cf. the derivation of (13.49)) we obtain the estimate

\[ \left|\overline{P}_{0i}(x^\delta,t)-\overline{P}_{0i}(x,t)\right| \leq (C)|\varphi|_\alpha \begin{cases} \delta^{\alpha'} & \text{for } i=1,\\ \delta^{\alpha^0}+\delta^{\beta'} & \text{for } i=2. \end{cases} \tag{14.42} \]

From (14.6), (14.24)—(14.26), (14.42), in view of (14.5) and Theorem 8, § 2 [1], (14.2) follows, which completes the proof of Theorem 18.

§ 15. PROOF OF THEOREM 19

From (14.5) and the following equalities

\[ \frac{\partial^2 p}{\partial x_1 \partial x_2} = -\,v_1 v_2^{-1}\frac{\partial^2 p}{\partial x_1^2} + v_2^{-1}\sum_{j=1}^{2}\left[N_j-(v,N)v_j\right]\, \frac{\partial^2 g_{0,\frac n2}}{\partial x_j \partial x_1}, \]

\[ \frac{\partial^2 p}{\partial x_2^2} = v_1^2 v_2^{-1}\frac{\partial^2 p}{\partial x_1^2} + \]

\[ + v_2^{-1}\sum_{j=1}^{2}\left[N_j-(v,N)v_j\right] \left[ \frac{\partial^2 g_{0,\frac n2}}{\partial x_j \partial x_2} - \frac{\partial^2 g_{0,\frac n2}}{\partial x_j \partial x_1} \right], \]

\[ \frac{\partial p}{\partial t} = \sum_{i=1}^{2}\frac{\partial^2 p}{\partial x_i^2}, \]

where \(v_j=v_j(\xi_1,\tau)\), \(N_j=N_j(\xi_1,\tau)\),

\[ p\equiv p(r(\xi,\overline{x}),t-\tau),\qquad g_{0,n/2}\equiv g_{0,n/2}(r(\xi,\overline{x}),t-\tau) \]

it follows that, by virtue of the maximum principle (cf. § 5 [1], § 9 [2], and § 14) and Theorem 11, § 7 [2], for the proof of Theorem 19 it is sufficient to show the validity of the relations (see Remark 12, § 12, and (14.3), (14.6)) under (10.31), (10.37), (12.15)

\[ \left|\overline{P}_0(\hat{x},t+\Delta t)-\overline{P}_0(x,t)\right| \leq (C)|\varphi|_{1+\alpha}|\Delta t|^{\frac{1+\alpha^0}{2}} \]

\[ \left|\overline{P}_0(\overline{x},t)\right| \leq (C)|\varphi|_{1+\alpha}t^{\frac{1+\alpha}{2}}, \tag{15.1} \]

\[ \left| \frac{\partial P_0(\overline{x},t)}{\partial \overline{x}_1} - \frac{\partial \overline{P}_0(x,t)}{\partial \overline{x}_1} \right| \leq (C)|\varphi|_{1+\alpha} r^{\alpha^0}(\overline{x},x) \tag{15.2} \]

(as \((\overline{x},t)\) approaches \((x,t)\in\Gamma_t\) along the normal \(N(x,t)\), which is the axis \(O\overline{\xi}_2\) in the local coordinate system \(\{\overline{\xi},\tau\}\) associated with \((x,t)\in\Gamma_t\)),

\[ \frac{\partial P_0(x,t)}{\partial x_1}\in H^{0,\alpha',\alpha'/2}(\Gamma), \]

\[ \left|\frac{\partial P_0(\bar x,t)}{\partial x_1}\right|\leq (C)|\varphi|_{1+\alpha}\,t^{\alpha/2},\qquad (\bar x,t)\in \overline{D_T^{\delta}} . \tag{15.3} \]

From (10.42) (for \(x=\bar x\)), (14.18), (14.16), (14.17), and (14.15) it follows that

\[ \begin{aligned} \varphi(\xi_1,\tau)\frac{\partial p(r(\xi,\bar x),t-\tau)}{\partial x_1} &= -\bigl(\varphi^*(\xi_1,\tau),q_1^{(3/2)}(r(\xi,\bar x),t-\tau)\bigr) \\ &= \bigl(\varphi^*(\xi_1,\tau),q_0^{(3/2)}(r(\xi,\bar x),t-\tau)\bigr) -\varphi_2^*(\xi_1,\tau)\times \\ &\quad \times \left[\frac{\partial\psi(x_1,t)}{\partial x_1} -\frac{\partial\psi(\xi_1,\tau)}{\partial \xi_1}\right] G_{3/2}^{(1)}(r(\xi,\bar x),t-\tau) \\ &\quad -\bigl(\varphi^*(\xi_1,\tau),r_{\xi\bar x}q_{11}^{(3/2)}(r(\xi,\bar x),t-\tau)\bigr), \end{aligned} \tag{15.4} \]

where the notations have been introduced (see (14.7), (14.8))

\[ q_{zz}^{(m)}(r(\xi,\bar x),t-\tau) = \frac{\partial G_m^{(z)}(r(\xi,\bar x),t-\tau)}{\partial x_1} + \frac{\partial G_m^{(z)}(r(\xi,\bar x),t-\tau)}{\partial \xi_1}, \qquad 0\leq z\leq 1, \tag{15.5} \]

and, in view of (13.23), from

\[ \begin{aligned} \frac{\partial}{\partial x_1}G_m^{(z)}(r(\xi,\bar x),t-\tau) &= -\frac12(\mu(z),r_{\xi\bar x})\times \\ &\quad \times \left[\left(\frac{\partial\mu(z)}{\partial x_1},r_{\xi\bar x}\right) +\left(\mu(z),\frac{\partial r_{\xi\bar x}}{\partial \xi_1}\right)\right] G_{m+1}^{(z)}(r(\xi,\bar x),t-\tau) \\ &\quad +\frac12\left[\left(\frac{\partial\nu(z)}{\partial x_1},r_{\xi\bar x}\right) +\left(\nu(z),\frac{\partial r_{\xi\bar x}}{\partial x_1}\right)\right] g_{0,m+\frac12}(r(\xi,\bar x),t-\tau), \end{aligned} \tag{15.6} \]

\[ \begin{aligned} \frac{\partial}{\partial \xi_1}G_m^{(z)}(r(\xi,\bar x),t-\tau) &= -\frac12(\mu(z),r_{\xi\bar x})\times \\ &\quad \times \left[\left(\frac{\partial\mu(z)}{\partial \xi_1},r_{\xi\bar x}\right) +\left(\mu(z),\frac{\partial r_{\xi\bar x}}{\partial \xi_1}\right)\right] G_{m+1}^{(z)}(r(\xi,\bar x),t-\tau) \\ &\quad +\frac12\left[\left(\frac{\partial\nu(z)}{\partial \xi_1},r_{\xi\bar x}\right) +\left(\nu(z),\frac{\partial r_{\xi\bar x}}{\partial \xi_1}\right)\right] g_{0,m+\frac12}(r(\xi,\bar x),t-\tau) \end{aligned} \tag{15.7} \]

it follows that

\[ q_{zz}^{(m)}(r(\xi,\bar x),t-\tau) = -\frac12(\mu(z),r_{\xi\bar x}) \left[\left(\frac{\partial\mu(z)}{\partial x_1} +\frac{\partial\mu(z)}{\partial \xi_1},r_{\xi\bar x}\right)+ \]

1. Differential Equations No. 11

\[ +\,\mu_2(z)\left(\frac{\partial\psi(x_1,\tau)}{\partial x_1} -\frac{\partial\psi(\xi_1,\tau)}{\partial \xi_1}\right) G_{m+1}^{(z)}(r(\xi,\bar x),t-\tau)+ \]

\[ +\frac12\left[\left(\frac{\partial\nu(z)}{\partial x_1} +\frac{\partial\nu(z)}{\partial \xi_1},\, r_{\bar\xi}\right)+\right. \tag{15.8} \]

\[ \left. +\nu_2(z)\left(\frac{\partial\psi(x_1,t)}{\partial x_1} -\frac{\partial\psi(\xi_1,\tau)}{\partial \xi_1}\right)\right] g_{0,m+\frac12}(r(\xi,\bar x),t-\tau), \]

where, in view of (14.7),

\[ \frac{\partial\nu(z)}{\partial x_1} =(1-z)\frac{\partial\gamma_0}{\partial x_1}\,\mu(z), \qquad \frac{\partial\mu(z)}{\partial x_1} =(z-1)\frac{\partial\gamma_0}{\partial x_1}\,\nu(z), \]

\[ \frac{\partial\nu(z)}{\partial \xi_1} =z\frac{\partial\gamma}{\partial \xi_1}\,\mu(z), \qquad \frac{\partial\mu(z)}{\partial \xi_1} =-z\frac{\partial\gamma}{\partial \xi_1}\,\nu(z), \]

and

\[ \frac{\partial\gamma_0}{\partial x_1} =\nu_1(x_1,\tau)\frac{\partial\nu_2(x_1,\tau)}{\partial x_1} - \]

\[ -\nu_2(x_1,\tau)\frac{\partial\nu_1(x_1,\tau)}{\partial x_1} =\left(\frac{\partial\nu(0)}{\partial x_1},\,\mu(0)\right), \]

\[ \frac{\partial\gamma}{\partial \xi_1} =\left(\frac{\partial\nu(1)}{\partial \xi_1},\,\mu(1)\right). \]

From (15.4) and (15.6), integrating by parts, which is justified as in § 5 [1] (see the derivation of (5.27)), and putting (see (14.14))

\[ \varphi^{*}_{i,j,k}(\xi_1,\tau) \equiv \varphi_i^{*}(\xi_1,\tau)(\mu_1(\xi_1,\tau))^j(\mu_2(\xi_1,\tau))^k \quad (i=0,1,2) \tag{15.9} \]

we obtain, by virtue of (12.5),

\[ P_0(\bar x,t) =-\sum_{k=1}^{2}\int_{t_1}^{t} \frac{\partial\varphi_k^{*}(\xi_1,\tau)}{\partial \xi_1} (x_1-\xi_1)^{2-k}(\bar x_2-\psi(\xi_1,\tau))^{k-1}\times \]

\[ \times G_{3/2}^{(1)}(r(\xi,\bar x),t-\tau)\,d\xi_1 -\int_{t_1}^{t}d\tau\int_{-\infty}^{+\infty} \varphi_2^{*}(\xi_1,\tau)\times \]

\[ \times\left[ \frac{\partial\psi(x_1,\tau)}{\partial x_1} -\frac{\partial\psi(\xi_1,\tau)}{\partial \xi_1} \right] G_{3/2}^{(1)}(r(\xi,\bar x),t-\tau)\,d\xi_1+ \]

\[ +\sum_{k=1}^{2}\frac12 C_2^k \int_{t_1}^{t}d\tau\int_{-\infty}^{+\infty} \varphi^{*}_{2,k,2-k}(\xi_1,\tau) \left[ \frac{\partial\psi(x_1,t)}{\partial x_1} -\frac{\partial\psi(\xi_1,\tau)}{\partial \xi_1} \right]\times \]

\[ \times(x_1-\xi_1)^k(\bar x_2-\psi(\xi_1,\tau))^{2-k} G_{5/2}^{(1)}(r(\xi,\bar x),t-\tau)\,d\xi_1+ \]

\[ +\sum_{k=0}^{2}\frac12 C_2^k \int_{t_1}^{t}d\tau\int_{-\infty}^{+\infty} \varphi^{*}_{0,k,2-k}(\xi_1,\tau) \frac{\partial\mu_1(\xi_1,\tau)}{\partial \xi_1} (x_1-\xi_1)^{k+1}\times \]

\[ \times(\bar x_2-\psi(\xi_1,\tau))^{2-k} G_{5/2}^{(1)}(r(\xi,\bar x),t-\tau)\,d\xi_1+ \]

\[ \begin{aligned} &+ \sum_{k=0}^{2}\frac{1}{2} C_2^k \int_{t_1}^{t} d\tau \int_{-\infty}^{+\infty} \varphi^*_{0,k,2-k}(\xi_1,\tau) \frac{\partial \mu_2(\xi_1,\tau)}{\partial \xi_1} (x_1-\xi_1)^k \times \\ &\qquad \times \bigl(\bar x_2-\psi(\xi_1,\tau)\bigr)^{3-k} G^{(1)}_{5/2}(r(\xi,\bar x),t-\tau)\,d\xi_1 - \\ &- \sum_{k=1}^{2}\frac{1}{2}\int_{t_1}^{t} d\tau \int_{-\infty}^{+\infty} \varphi^*_{0,2-k,k}(\xi_1,\tau)\nu_2(\xi_1,\tau) \left( \frac{\partial \psi(x_1,t)}{\partial x_1} -\frac{\partial \psi(\xi_1,\tau)}{d\xi_1} \right) (x_1-\xi_1)^{2-k} \bigl(\bar x_2-\psi(\xi_1,\tau)\bigr)^{k-1} \\ &\qquad \times g_{0,2}(r(\xi,\bar x),t-\tau)\,d\xi_1 - \\ &- \sum_{k=1}^{2}\frac{1}{2}\int_{t_1}^{t} d\tau \int_{-\infty}^{+\infty} \varphi^*_{0,2-k,k}(\xi_1,\tau) \frac{\partial \nu_1(\xi_1,\tau)}{\partial \xi_1} (x_1-\xi_1)^{3-k} \times \\ &\qquad \times \bigl(\bar x_2-\psi(\xi_1,\tau)\bigr)^{k-1} g_{0,2}(r(\xi,\bar x),t-\tau)\,d\xi_1 - \\ &- \sum_{k=1}^{2}\frac{1}{2}\int_{t_1}^{t} d\tau \int_{-\infty}^{+\infty} \varphi^*_{0,2-k,k}(\xi_1,\tau) \frac{\partial \nu_2(\xi_1,\tau)}{\partial \xi_1} (x_1-\xi_1)^{2-k} \times \\ &\qquad \times \bigl(\bar x_2-\psi(\xi_1,\tau)\bigr)^k g_{0,2}(r(\xi,\bar x),t-\tau)\,d\xi_1 = \sum_{i=1}^{17} P_{0i}(\bar x,t). \end{aligned} \tag{15.10} \]

From the estimates (see (11.10))

\[ \left| \frac{\partial \varphi_k^*(\xi_1,\tau)}{\partial \xi_1} \right| \leq (C)|\varphi|_{1+\alpha}\tau^{-\frac{\alpha}{2}}, \qquad |\varphi_0^*(\xi_1,\tau)| \leq (C)|\varphi|_{1+\alpha}\tau^{\frac{1+\alpha}{2}} \tag{15.11} \]

with the aid of Lemma 11 of § 10 and Lemma 4 of § 1 [1], as well as (6.5) of § 6 [2], we obtain (cf. the derivation of (5.25) in § 5 [1])

\[ |P_{0i}(\bar x,t)| \leq (C)|\varphi|_{1+\alpha}\,t^{\frac{1+\alpha}{2}}, \quad (\bar x,t)\in \bar D_T^\delta \quad (i=1,2,\ldots,17). \tag{15.12} \]

Let us now consider the direct values \(\bar P_{0i}(x_1,t)\); for this we pass in (15.10) to the limit as \(\delta\to 0\). Then, in the usual way, we obtain

\[ \left| \bar P_{0i}(x_1,t+\Delta t)-\bar P_{0i}(x_1,t) \right| \leq (C)|\varphi|_{1+\alpha}|\Delta t|^{\frac{1+\alpha^0}{2}} \tag{15.13} \]

\[ (i=2,4,6,7,9\text{--}11,13,15\text{--}17). \]

For \(i=1,3,5,8,12,14\), set

\[ \bar P_{0i}(x_1,t)=\bar P_{0i}^{(0)}(x_1,t)+\bar P_{0i}^{(1)}(x_1,t), \]

where (cf. (13.13), (13.14), (13.16)—(13.19))

\[ \bar P_{01}^{(0)}(x_1,t) = \int_{t_1}^{t} \frac{\partial \varphi_1^*(x_1,\tau)}{\partial x_1}\,d\tau \int_{-\infty}^{+\infty} \xi_1 G_{0,3/2}(\xi_1,t-\tau)\,d\xi_1 = \]

\[ = -4\int_{t_1}^{t} (t-\tau)^{-\frac12} \frac{\partial \varphi_1^*(x_1,\tau)}{\partial x_1} v_2^{-2}(x_1,\tau)\Phi_0(\tau)\,d\tau, \]

\[ \bar P_{03}^{(0)}(x_1,t) = -\int_{t_1}^{t} \varphi_2^*(x_1,\tau) \left[ \frac{\partial \psi(x_1,t)}{\partial x_1} - \frac{\partial \psi(x_1,\tau)}{\partial x_1} \right]d\tau \times \]

\[ \times \int_{-\infty}^{+\infty} G_{0,3/2}(\xi_1,t-\tau)\,d\xi_1 = \pi\int_{t_1}^{t} (t-\tau)^{-1}\varphi_2^*(x_1,\tau)v_2^{-1}(x_1,\tau) \times \]

\[ \times \left[ \frac{\partial \psi(x_1,t)}{\partial x_1} - \frac{\partial \psi(x_1,\tau)}{\partial x_1} \right]d\tau, \]

\[ \bar P_{05}^{(0)}(x_1,t) = \frac12\int_{t_1}^{t} \varphi_{2,2,0}^*(x_1,\tau) \left[ \frac{\partial \psi(x_1,t)}{\partial x_1} - \frac{\partial \psi(x_1,\tau)}{\partial x_1} \right]d\tau \times \]

\[ \times \int_{-\infty}^{+\infty} \xi_1^2G_{0,5/2}(\xi_1,t-\tau)\,d\xi_1 = -\pi\int_{t_1}^{t} (t-\tau)^{-1}\varphi_{2,2,0}^*(x_1,\tau) \times \]

\[ \times \left[ \frac{\partial \psi(x_1,t)}{\partial x_1} - \frac{\partial \psi(x_1,\tau)}{\partial x_1} \right] v_2^{-3}(x_1,\tau)\Phi_1(\tau)\,d\tau, \]

\[ \bar P_{08}^{(0)}(x_1,t) = -\frac12\int_{t_1}^{t} \varphi_{0,2,0}^*(x_1,\tau) \frac{\partial \mu_1(x_1,\tau)}{\partial x_1}\,d\tau \times \]

\[ \times \int_{-\infty}^{+\infty} \xi_1^3G_{0,5/2}(\xi_1,t-\tau)\,d\xi_1 = 8\int_{t_1}^{t} (t-\tau)^{-\frac12}\varphi_{0,2,0}^*(x_1,\tau) \times \]

\[ \times \frac{\partial \mu_1(x_1,\tau)}{\partial x_1} v_2^{-4}(x_1,\tau)\Phi_1(\tau)\,d\tau, \]

\[ \bar P_{012}^{(0)}(x_1,t) = \frac12\int_{t_1}^{t} \varphi_{0,1,1}^*(x_1,\tau)v_2(x_1,\tau) \left[ \frac{\partial \psi(x_1,t)}{\partial x_1} - \right. \]

\[ \left. - \frac{\partial \psi(x_1,\tau)}{\partial x_1} \right]d\tau \int_{-\infty}^{+\infty} \xi_1 g_{0,2}(\xi_1,t-\tau)\,d\xi_1 = 0, \]

\[ \overline{P}_{014}^{(0)}(x_1,t)=-\frac{1}{2}\int_{t_1}^{t}\varphi_{0,1,1}(x_1,\tau)\frac{\partial v_1(x_1,\tau)}{\partial x_1}\,d\tau \int_{-\infty}^{+\infty}\xi_1^2 g_{0,2}(\xi_1,t-\tau)\,d\xi_1 = \]

\[ =-2\sqrt{\pi}\int_{t_1}^{t}(t-\tau)^{-1/2}\varphi_{0,1,1}(x_1,\tau)\frac{\partial v_1(x_1,\tau)}{\partial x_1}\,d\tau . \]

By virtue of Lemma 3 of § 1 [1],

\[ \left|\overline{P}_{0i}^{(0)}(x_1,t+\Delta t)-\overline{P}_{0i}^{(0)}(x_1,t)\right| \leq (C)|\varphi|_{1+\alpha}|\Delta t|^{\frac{1+\alpha^0}{2}} \tag{15.14} \]

\[ (i=1,\ 8,\ 14). \]

If, however, one uses inequality (6.4), [2], then (cf. the derivation of (13.26)) it is easy to show that (15.14) is also valid for \(i=3,5\). From Lemma 11 of § 10 and Lemma 13 of § 13 we obtain

\[ \left|\overline{P}_{0i}^{(1)}(x_1,t)\right| \leq (C)|\varphi|_{1+\alpha}t^{\frac{1+\alpha}{2}} \qquad (i=1,2,\ldots,17). \]

With the aid of Lemma 7 of § 6 [2], Lemma 12 of § 10, Lemma 13 of § 13, the estimates (6.3)—(6.5), (6.11), (6.12) of § 6 [2], (1.23), (1.24) of § 1 [1], in the usual way (cf. the derivation of (13.31)) we obtain (15.14) for \(\overline{P}_{0i}^{(1)}\), \(i=1,2,\ldots,17\), whence (15.1) follows.

From (14.6), (15.4), integrating by parts, we obtain (which is justified as in § 5 [1]) the representation (cf. (14.15), (14.24), (14.25))

\[ \begin{aligned} P_{00}(\overline{x},t)\equiv \frac{\partial P_0(\overline{x},t)}{\partial x_1} &= \int_{t_1}^{t}d\tau\int_{-\infty}^{+\infty} \left(\frac{\partial\varphi^*(x_1,\tau)}{\partial x_1}, q_0^{(3/2)}(r(\xi,\overline{x}),t-\tau)\right)d\xi_1 \\ &\quad +\int_{t_1}^{t}d\tau\int_{-\infty}^{+\infty} \left(-\frac{\partial\varphi^*(x_1,\tau)}{\partial x_1} -\frac{\partial\varphi^*(\xi_1,\tau)}{\partial \xi_1}, q_i^{(3/2)}(r(\xi,\overline{x}),t-\tau)\right)d\xi_1 \\ &\quad -\int_{t_1}^{t}d\tau\int_{-\infty}^{+\infty} \left(-\frac{\partial\varphi^*(x_1,\tau)}{\partial x_1}, q_{10}^{(3/2)}(r(\xi,\overline{x}),t-\tau)\right)d\xi_1 \\ &\quad -\int_{t_1}^{t}d\tau\int_{-\infty}^{+\infty} \Bigg[ \frac{\partial\varphi_2^*(\xi_1,\tau)}{\partial \xi_1} \left(\frac{\partial\psi(x_1,t)}{\partial x_1} -\frac{\partial\psi(\xi_1,\tau)}{\partial \xi_1}\right) \\ &\qquad\qquad +\varphi_2^*(\xi_1,\tau) \left(\frac{\partial^2\psi(x_1,t)}{\partial x_1^2} -\frac{\partial^2\psi(\xi_1,\tau)}{\partial \xi_1^2}\right) \Bigg] G_{3/2}^{(1)}(r(\xi,\overline{x}),t-\tau)\,d\xi_1 \\ &\quad -\int_{t_1}^{t}d\tau\int_{-\infty}^{+\infty} \varphi_2^*(\xi_1,t) \left(\frac{\partial\psi(x_1,t)}{\partial x_1} -\frac{\partial\psi(\xi_1,\tau)}{\partial \xi_1}\right) q_{11}^{(3/2)}(r(\xi,\overline{x}),t-\tau)\,d\xi_1 \\ &\quad -\int_{t_1}^{t}d\tau\int_{-\infty}^{+\infty} \left(\frac{\partial\varphi^*(\xi_1,\tau)}{\partial \xi_1}, r_{\xi x}\,q_{00}^{(3/2)}(r(\xi,\overline{x}),t-\tau)\right)d\xi_1 - \end{aligned} \]

\[ -\int_{t_1}^{t} d\tau \int_{-\infty}^{+\infty} \left(\varphi^{*}(\xi_1,\tau),\, q_{1100}^{(3/2)}(r(\xi,\bar{x}),t-\tau)\right)\,d\xi_1 \equiv \sum_{i=0}^{6} P_{00}^{(i)}(\bar{x},t); \tag{15.15} \]

where the notation (14.4), (14.16), (14.17), (15.5) has been used, and

\[ q_{1100}^{(m)}(r(\xi,\bar{x}),t-\tau) = \frac{\partial}{\partial x_1} \left[ r_{\xi x}\,q_{11}^{(m)}(r(\xi,\bar{x}),t-\tau) \right] + \frac{\partial}{\partial \xi_1} \left[ r_{\xi x}\,q_{00}^{(m)}(r(\xi,\bar{x}),t-\tau) \right], \tag{15.16} \]

obviously, for \(\delta>0\),

\[ P_{00}^{(0)}(\bar{x},t)\equiv 0. \tag{15.17} \]

Analogously to Lemmas 14–16, the following Lemmas 17–19 are proved.

Lemma 17. If \(\Gamma\) is of type \(\Pi_{1,1,\frac{1+\alpha}{2}}^{1,\alpha,\frac{\alpha}{2}}\) and \(v\in H_{1,\beta,\frac{\beta}{2}}^{0,1,\frac{1+\beta}{2}}(\Gamma)\) \((0<\alpha\leqslant\beta\leqslant1)\), then under (12.15), (14.14) the estimates (14.20), (14.21) (where \(\beta=1\)), (14.22), (14.23) (where \(\alpha=1\)) of Lemma 14 § 14 hold, and also (see (15.5), (15.16))

\[ \left|q_{zz}^{(m)}(r(\xi,\bar{x}),t-\tau)\right| \leqslant (C)\,g_{0,m}(\varkappa r(\xi,\bar{x}),2(t-\tau)),\qquad 0\leqslant z\leqslant 1, \tag{15.18} \]

\[ \left|q_{1100}^{(m)}(r(\xi,\bar{x}),t-\tau)\right| \leqslant (C)\,g_{0,m-\beta/2}(\varkappa r(\xi,\bar{x}),2(t-\tau)), \tag{15.19} \]

\[ \left|q_{zz}^{(m)}(r(\xi,\bar{x}),t-\tau) -\bar{q}_{zz}^{(m)}(r(\xi,x),t-\tau)\right| \leqslant (C)\delta\,g_{0,m+1/2}(\varkappa\rho,2(t-\tau))\times \]

\[ \times \int_{0}^{1} g_{00}\bigl(\varkappa|\theta\delta-\psi(\xi_1,\tau)|,2(t-\tau)\bigr)\,d\theta, \qquad 0\leqslant z\leqslant 1, \]

\[ \left|q_{1100}^{(m)}(r(\xi,\bar{x}),t-\tau) -\bar{q}^{(m)}(r(\xi,x),t-\tau)\right| \leqslant (C)\delta\,g_{0,m+\frac{1-\alpha}{2}}(\varkappa\rho,2(t-\tau))\times \]

\[ \times \int_{0}^{1} g_{0,0}\bigl(\varkappa|\theta\delta-\psi(\xi_1,\tau)|,2(t-\tau)\bigr)\,d\theta. \]

Lemma 18. If for \(\Gamma\) and \(v\) the hypotheses of Lemma 17 are satisfied, then under (10.31)

\[ \left| \bar{q}_{1}^{(m)}(r(\xi,\hat{x}),t+\Delta t-\tau) - \bar{q}_{1}^{(m)}(r(\xi,x),t-\tau) \right| \leqslant \]

\[ \leqslant (C)\left[ (\Delta t)^{\frac{1+\alpha}{2}}(t-\tau)^{-m} + \Delta t\,(t-\tau)^{-m-1} \right] g_{0,0}(\varkappa\rho,2(t+\Delta t-\tau)), \]

\[ \left| \bar{q}_{10}^{(m)}(r(\xi,x),t+\Delta t-\tau) - \bar{q}_{10}^{(m)}(r(\xi,x),t-\tau) \right| \leqslant \]

\[ \leq (C)\left[(\Delta t)^{\frac{1+\alpha}{2}}(t-\tau)^{-m+\frac12} +\Delta t\,(t-\tau)^{-m-\frac12}\right]g_{0,0}\bigl(\varkappa\rho,\,2(t+\Delta t-\tau)\bigr), \]

\[ \left|\bar q_{zz}^{(m)}(r(\xi,\hat x),\,t+\Delta t-\tau) -\bar q_{zz}^{(m)}(r(\xi,x),\,t-\tau)\right|\leq \]

\[ \leq (C)\left[(\Delta t)^{\frac{1+\alpha}{2}}(t-\tau)^{-m-\frac12} +\Delta t\,(t-\tau)^{-m-1}\right] g_{0,0}\bigl(\varkappa\rho,\,2(t+\Delta t-\tau)\bigr), \]

\[ 0\leq z\leq 1, \]

\[ \left|\bar q_{1100}^{(m)}(r(\xi,\hat x),\,t+\Delta t-\tau) -\bar q_{1100}^{(m)}(r(\xi,x),\,t-\tau)\right|\leq \]

\[ \leq (C)\left[((\Delta t)^{\frac{\alpha}{2}}+(\Delta t)^{\frac{1+\alpha}{2}})(t-\tau)^{-m} +\Delta t\,(t-\tau)^{-m-1+\frac{\alpha}{2}}\right] g_{0,0}\bigl(\varkappa\rho,\,2(t+\Delta t-\tau)\bigr). \]

Lemma 19. If, for \(\Gamma\) and \(v\), the hypotheses of Lemma 17 are satisfied, then under (10.37), (12.4), (12.11) the estimates (14.20), (14.21) \((\beta=1)\), (15.18), (15.19) (for \(x\equiv x\)) of Lemma 17, (14.33), (14.34) \((\beta=1)\), (14.35) of Lemma 16 hold, and also

\[ \left|\bar q_{zz}^{(m)}(r(\xi,x^\delta),\,t-\tau)\right| \leq (C)g_{0m}\bigl(\varkappa\rho_1,\,2(t-\tau)\bigr),\qquad 0\leq z\leq 1, \]

\[ \left|\bar q_{1100}^{(m)}(r(\xi,x^\delta),\,t-\tau)\right| \leq (C)g_{0,m-\frac{\beta}{2}}\bigl(\varkappa\rho_1,\,2(t-\tau)\bigr) \]

and, for \(\rho\geq 2\delta\) (see (12.13)),

\[ \left|\bar q_{10}^{(m)}(r(\xi,x^\delta),\,t-\tau) -\bar q_{10}^{(m)}(r(\xi,x),\,t-\tau)\right|\leq \]

\[ \leq (C)\sum_{k=1}^{2}\delta^k g_{0,m+\frac{k-1}{2}}\left(\frac{\varkappa\rho}{2},\,2(t-\tau)\right), \]

\[ \left|\bar q_{zz}^{(m)}(r(\xi,x^\delta),\,t-\tau) -\bar q_{zz}^{(m)}(r(\xi,x),\,t-\tau)\right|\leq \]

\[ \leq (C)\sum_{k=1}^{2}\delta^k g_{0,m+\frac{k}{2}}\left(\frac{\varkappa\rho}{2},\,2(t-\tau)\right), \]

\[ \left|\bar q_{1100}^{(m)}(r(\xi,x^\delta),\,t-\tau) -\bar q_{1100}^{(m)}(r(\xi,x),\,t-\tau)\right|\leq \]

\[ \leq (C)\left[ \sum_{k=0}^{1}\delta^{\alpha+k} g_{0,m+\frac{k}{2}}\left(\frac{\varkappa\rho}{2},\,2(t-\tau)\right) +\delta\,g_{0,m+\frac{1-\alpha}{2}}\left(\frac{\varkappa\rho}{2},\,2(t-\tau)\right) +\sum_{k=1}^{3}\delta^{k+1} g_{0,m+\frac{k}{2}}\left(\frac{\varkappa\rho}{2},\,2(t-\tau)\right) \right]. \]

From Lemmas 11, 12 of § 10, Lemmas 17–19, estimates (6.3)—(6.5), (6.11), (6.12) of § 6 [2], (15.12), condition (13.23), representation (15.15), and identity (15.17), we obtain without difficulty (cf. the considerations of § 14) (15.2), (15.3), which completes the proof of Theorem 19. It is also easy to verify the validity of Remark 11 of § 11.

§ 16. PROOF OF THEOREM 20

Without loss of generality we shall assume that the functions \(f_1\) and \(f_2\) from (11.12), (11.13) have been extended with preservation of the smoothness class to \(\overline{D}_T\) and \(\overline{\Omega}=\overline{D}_T\cap\{t=0\}\), respectively.

Lemma 20 (cf. [9]). Let \(f_1(\bar x,t)\) be defined on \(\overline{D}_T\), and let

\[ f_1\in H^{0,\alpha,\frac{\alpha}{2}}(\overline{D}_T). \]

Then there exists a solution \(u_1(\bar x,t)\) of equation (11.12), with \(u_1(\bar x,0)=0\), \(\bar x\in\overline{\Omega}\), and

\[ u_1\in H^{1,\alpha,\frac{\alpha}{2}}_{1,1,\frac{1+\alpha}{2}}(\overline{D}_T), \qquad |u_1|_{2+\alpha}\le (C)|f_1|_\alpha . \]

Lemma 21 (cf. [9]). Let \(f_2(\bar x)\) be defined on \(\overline{\Omega}\) and \(|f_2|_{2+\alpha}<+\infty\). Then there exists a solution \(u_2(\bar x,t)\) of the homogeneous equation (11.12) (where \(f_1\equiv0\)), with \(u_2(\bar x,0)=f_2(\bar x)\), \(\bar x\in\overline{\Omega}\), and

\[ u_2\in H^{1,\alpha,\frac{\alpha}{2}}_{1,1,\frac{1+\alpha}{2}}(\overline{D}_T), \qquad |u_2|_{2+\alpha}\le (C)|f_2|_{2+\alpha}. \]

From Lemmas 20, 21 it follows that it is sufficient to prove Theorem 20 under the assumption \(f_1(\bar x,t)\equiv f_2(\bar x)\equiv0\) (problem \((11.12^\circ),(11.13^\circ),(11.14)\)). The solution of problem \((11.12^\circ),(11.13^\circ),(11.14)\) is sought in the form of the special heat potential (10.43) with unknown density \(\varphi(y,\tau)\). Use of the boundary condition (11.14), in view of the jump formula (11.4) of Theorem 16, § 11, leads to the Volterra integral equation of the second kind (see (10.43), (10.44))

\[ \frac{(2\sqrt{\pi})^n}{2}\,\varphi(x,t) = -f_3(x,t)+\overline{Q}(x,t)+b(x,t)\overline{P}(x,t) \tag{16.1} \]

\[ (x,t)\in\Gamma \]

for determining the density \(\varphi(x,t)\). Since the kernels of equation (16.1) have weak singularities (see § 6 [7]), this equation has a unique solution \(\varphi(x,t)\) continuous on \(\Gamma\). By Theorems 14, 15 of § 11 and the condition

\[ b,\ f_3\in H^{0,1,\frac{1+\alpha}{2}}_{1,\alpha,\alpha/2}(\Gamma), \]

the density \(\varphi\), determined by equality (16.1), belongs to the class \(H^{0,\beta,\beta/2}(\Gamma)\), and by Theorems 17, 18 of § 11 one may assert that

\[ \varphi\in H^{0,1,\frac{1+\alpha}{2}}_{1,\alpha,\alpha/2}(\Gamma). \]

But then Remark 11 to Theorem 19 of § 11 shows that

\[ u(\bar x,t)\in H^{1,\alpha,\alpha/2}_{1,1,\frac{1+\alpha}{2}}(\overline{D}_T^\delta). \]

Theorem 20 is thereby proved.

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Received by the editors
July 2, 1965

Moscow State University
named after M. V. Lomonosov