THE FIRST BOUNDARY VALUE PROBLEM FOR QUASILINEAR PARABOLIC EQUATIONS WITH RETARDED ARGUMENT

V. V. Podgornov

The question of the existence of a solution of the equation

\[ \frac{\partial u(t,x)}{\partial t} = \sum_{i,j=1}^{n} \frac{\partial}{\partial x_i} \left( a_{ij}(t,x,u(t,x))\, \frac{\partial u(t,x)}{\partial x_j} \right) + \]

\[ + a\left(t,x,u(t,x),\ldots,u(g_r(t),x),\ldots, \frac{\partial u(t,x)}{\partial x_k},\ldots\right) \tag{1} \]

in the cylinder \(Q=\Omega\times[0,T]\) is considered, under the boundary conditions

\[ u(t,x)\big|_{S}=0,\qquad u(t,x)\big|_{t\le 0}=0. \tag{2} \]

Here \(x=(x_1,\ldots,x_n)\); \(k=1,\ldots,n\); \(g_r(t)\le t\); \(r=1,\ldots,q\); \(\Omega\) is the base of the cylinder; \(S\) is its lateral surface.

For brevity, let us rewrite equation (1) in the form

\[ \frac{\partial u}{\partial t} = \sum_{i,j=1}^{n} \frac{\partial}{\partial x_i} \left( a_{ij}(t,x,u)\, \frac{\partial u}{\partial x_j} \right) + a\bigl(t,x,u,u(g_r),u_{x_k}\bigr). \]

Let \(\Gamma\) denote the union of the set of points \((t,x)\) for which \(t\le 0,\ x\in\Omega\), and the set of points \((t,x)\) lying on the lateral surface of the cylinder; let \(\mu(\tau)\) be a positive monotonically nondecreasing function, and \(\nu(\tau)\) a positive monotonically nonincreasing function of \(\tau\ge 0\).

Introduce the notation:

\[ |u|_0=\max_{\overline Q}|u(t,x)|;\qquad p=\sqrt{\sum_{k=1}^{n}p_k^2};\qquad |\nabla u|=\sqrt{\sum_{k=1}^{n}u_{x_k}^{2}}. \]

Let \(C^{\alpha,\alpha/2}(\overline Q)\) denote the class of functions \(u(t,x)\) continuous in \(\overline Q\) for which the Hölder norm is finite,

\[ |u|_{\alpha,Q} = |u|_0 + \max_{(t,x),(t',x')\in Q} \frac{|u(t,x)-u(t',x')|} {\left(|t-t'|+|x-x'|^2\right)^{\alpha/2}}, \]

and let \(C_{2,1}^{\alpha,\alpha/2}(\overline Q)\) denote the set of functions \(u(t,x)\) with derivatives \(u_{x_i x_j}\), \(i,j=1,\ldots,n\), and \(u_t\) continuous in \(\overline Q\), which satisfy in \(\overline Q\) a Hölder condition with exponent \(\alpha\) in \(x\) and \(\alpha/2\) in \(t\) \((\alpha\in(0,1))\).

The norm in \(C_{2,1}^{\alpha,\alpha/2}(\overline Q)\) is defined by the equality

\[ |u|_{C^{\alpha,\alpha/2}_{2,1}(\overline Q)} = |u|_0+\sum_{i,j=1}^n |u_{x_i x_j}|_{\alpha,Q}+|u_t|_{\alpha,Q}. \]

We shall say that a domain \(\Omega \in A_2^\alpha\) if its boundary can be divided into a finite number of pieces, each of which can be represented in the form

\[ x_i=\psi(x_1,\ldots,x_{i-1},x_{i+1},\ldots,x_n) \]

for some \(i\), and the function \(\psi\in C_2^\alpha\) in the corresponding domain of variation of the arguments. Here \(C_2^\alpha\) is the class of twice continuously differentiable functions satisfying the Hölder condition with exponent \(\alpha\). The following is valid.

Theorem 1. Suppose that

1) for \((t,x)\in \overline Q\) and arbitrary \(u,u(g_r)\) the inequalities

\[ \nu(|u|)\sum_{i=1}^n \xi_i^2 \le \sum_{i,j=1}^n a_{ij}(t,x,u)\xi_i\xi_j \le \mu(|u|)\sum_{i=1}^n \xi_i^2, \tag{3} \]

\[ a_u(t,x,u,u(g_r),0)\le N;\qquad 0\le a_{u(g_l)}(t,x,0,u(g_r),0)\le N_l; \]

\[ |a(t,x,0,0,0)|\le N_0;\qquad r,l=1,\ldots,q;\quad N,N_l,N_0\text{ are certain constants;} \]

2) for \((t,x)\in \overline Q\), \(|u|\le e^{kT}=M\), \(|u(g_r)|\le M\) \(\bigl(k>|N|+\sum_{r=1}^q N_r+N_0\bigr)\), and arbitrary \(p_k\), the functions \(a_{ij}(t,x,u)\) and \(a(t,x,u(g_r),p_k)\) are continuous, the functions \(a_{ij}\) are differentiable with respect to \(x_k,u\), and the function \(a\) satisfies the inequality

\[ |a(t,x,u,u(g_r),p_k)|\le \mu_1(1+p)^2,\qquad \mu_1=\mathrm{const}>0; \tag{4} \]

3) in each finite part of the domain

\[ H:\left\{(t,x)\in\overline Q,\ |u|\le M,\ |u(g_r)|\le M,\ -\infty<p_k<\infty\right\} \]

the functions \(a_{ij}\), \(\dfrac{\partial a_{ij}}{\partial x_k}\), \(\dfrac{\partial a_{ij}}{\partial u}\), \(a\) are continuous functions satisfying, in \(t,x,u,u(g_r),p_k\), the Hölder condition with exponents \(\beta/2,\beta,\beta,\beta,\beta\), respectively;

4) the functions \(a_{ij}(t,x,0)\) for \((t,x)\in S\) satisfy the Lipschitz condition;

5) \(a(0,x,0,0,0)=0\) for \(x\in\sigma\) (\(\sigma\) is the boundary of \(\Omega\));

6) the functions \(g_r(t)\) on the interval \([0,T]\) satisfy the Hölder condition with exponent \(\beta\);

7) \(\Omega\in A_2^\beta\).

Under these conditions there exists a solution of problem (1), (2) belonging to the class \(C^{\beta^2,\beta^2/2}_{2,1}(\overline Q)\). This solution is unique if, in addition, the condition

8) in each finite part of the domain \(H\) the functions \(\dfrac{\partial a_{ij}}{\partial x_k}\), \(\dfrac{\partial a_{ij}}{\partial u}\), \(a\) are differentiable with respect to \(u,u(g_r),p_k\)

is fulfilled.

To prove the existence of a solution of problem (1), (2), it is enough to obtain a priori estimates of the following quantities:

\[ |u|_0,\quad |u|_{\alpha,Q},\quad \max_{\overline Q}|\nabla u|,\quad |u_{x_i}|_{\alpha,Q},\qquad i=1,\ldots,n. \]

First we obtain an estimate of \(|u|_0\), for which we shall need

Lemma. Let the function \(u(t,x)\), continuous in \(Q\), satisfy in \(Q\) the equation

\[ u_t=\sum_{i,j=1}^{n} a_{ij}(t,x)u_{x_i x_j} +\sum_{i=1}^{n} b_i(t,x)u_{x_i} + \]

\[ +\sum_{r=1}^{q} c_r(t,x)u(g_r)+c(t,x)u+f(t,x), \]

where the coefficients \(a_{ij}(t,x)\) satisfy the inequality

\[ \sum_{i,j=1}^{n} a_{ij}(t,x)\xi_i\xi_j \geq \nu \sum_{i=1}^{n}\xi_i^2,\quad \nu=\operatorname{const}>0, \]

and the functions \(c_r(t,x)\geq 0,\ r=1,\ldots,q\). Suppose, further, that the function \(z(t,x)\), continuous in \(Q\), satisfies in \(Q\) the inequalities

\[ z_t>\sum_{i,j=1}^{n} a_{ij}(t,x)z_{x_i x_j} +\sum_{i=1}^{n} b_i(t,x)z_{x_i} + \]

\[ +\sum_{r=1}^{q} c_r(t,x)z(g_r)+c(t,x)z+f(t,x), \]

\[ z_t<\sum_{i,j=1}^{n} a_{ij}(t,x)z_{x_i x_j} +\sum_{i=1}^{n} b_i(t,x)z_{x_i} + \]

\[ +\sum_{r=1}^{q} c_r(t,x)z(g_r)+c(t,x)z+f(t,x) \]

and the inequalities

\[ z(t,x)\big|_{\Gamma}>u(t,x)\big|_{\Gamma}, \tag{5} \]

\[ z(t,x)\big|_{\Gamma}<u(t,x)\big|_{\Gamma}, \]

then everywhere in \(\overline{Q}\)

\[ z(t,x)>u(t,x), \]

\[ z(t,x)<u(t,x). \tag{6} \]

Proof. Suppose the contrary, i.e., let there be at least one point \((t^0,x^0)\) at which \(z(t^0,x^0)=u(t^0,x^0)\). Consider the function
\(\eta(t,x)=z(t,x)-u(t,x)\). Let \(D\) be the set of points \((t,x)\) for which \(\eta(t,x)=0\). The set \(D\) is closed and, by condition (5), is located inside \(Q\). The function \(\eta(t,x)\) satisfies the equation

\[ \eta_t=\sum_{i,j=1}^{n} a_{ij}(t,x)\eta_{x_i x_j} +\sum_{i=1}^{n} b_i(t,x)\eta_{x_i} + \]

\[ +\sum_{r=1}^{q} c_r(t,x)\eta(g_r)+c(t,x)\eta+\varphi(t,x), \quad \varphi(t,x)>0. \tag{7} \]

Let the point \((t',x')\in D\) be such that its distance to the base of the cylinder is equal to the distance between the set \(D\) and the domain \(\Omega\). Consider equation (7) at the point \((t',x')\). We have

\[ \sum_{i,j=1}^{n} a_{ij}\eta_{x_i x_j} \geq 0,\quad \eta_t \leq 0,\quad \eta_{x_i}=0,\quad \eta=0, \]

\[ \sum_{r=1}^{q} c_r(t,x)\eta(g_r) \geq 0, \]

i.e., the left-hand side of equality (7) is negative or equal to zero, while the right-hand side is positive. The contradiction obtained proves inequality (6). The proof of the opposite inequality is analogous.

By virtue of the lemma, the following estimate is valid for the solution \(u(t,x)\) of problem (1), (2):

\[ |u|_{\bar Q} \leq e^{kT},\quad k>|N|+\sum_{r=1}^{q}N_r+N_0 . \tag{8} \]

Indeed, write equation (1) in the form (here and below, summation over two identical indices is assumed)

\[ u_t=a_{ij}(t,x,u)u_{x_i x_j} +\left(\frac{\partial a_{ij}}{\partial x_i} +\frac{\partial a_{ij}}{\partial u}u_{x_i}+b_j\right)u_{x_j}+ \]

\[ {}+d_lu(g_l)+du+d_0,\quad i,j=1,\ldots,n;\quad l=1,\ldots,q. \]

Here

\[ b_j=\int_0^1 \frac{\partial}{\partial u_{x_j}} \,a(t,x,u,u(g_r),\tau u_{x_k})\,d\tau;\quad d=\int_0^1 \frac{\partial}{\partial u} \,a(t,x,\tau u,u(g_r),0)\,d\tau; \]

\[ d_l=\int_0^1 \frac{\partial}{\partial u(g_l)} \,a(t,x,0,\tau u(g_r),0)\,d\tau;\quad d_0=a(t,x,0,0,0). \]

The function \(z=e^{kt}\), for \(t\in[0,T]\), \(k>|N|+\sum_{r=1}^{q}N_r+N_0\), satisfies the inequalities

\[ z_t>a_{ij}(t,x,u)z_{x_i x_j} +\left(\frac{\partial a_{ij}}{\partial x_i} +\frac{\partial a_{ij}}{\partial u}u_{x_i}+b_i\right)z_{x_j} + \]

\[ {}+d_l z(g_l)+dz+d_0,\quad z|_{\Gamma}>0. \]

Consequently, by virtue of the lemma we have

\[ u(t,x)<e^{kt}\leq e^{kT}=m,\quad (t,x)\in \bar Q. \tag{9} \]

Analogously, using the function \(\bar z=-e^{kt}\), we find

\[ u(t,x)>-e^{kT}\geq -e^{kT}=-m. \tag{10} \]

Estimate (8) follows from estimates (9), (10).

Estimates of the quantities \(|u|_{a,Q}\) and \(\max_{\bar Q}|\nabla u|\) by constants depending only on the number \(m\), the numbers \(\mu(m)\), \(\nu(m)\), \(\mu_1\) from inequalities (3), (4), and the maxima of the moduli of the second derivatives of the functions defining the boundary of the domain \(\Omega\), are obtained analogously to how this is done in [1]. We note that for

to obtain the estimate \(\max |\nabla u|\) in the whole domain \(Q\), an estimate of \(|\nabla u|_S\) is required. The estimate of \(|\nabla u|_S\) can be obtained in the same way as was done in [3] (pp. 394–397).

After the estimate \(\max\limits_{\overline Q} |\nabla u|\) has been obtained, the solution \(u(t,x)\) of equation (1) may be regarded as the solution of the linear equation

\[ u_t=a_{ij}(t,x)u_{x_i x_j}+b(t,x). \]

Here

\[ b(t,x)=\frac{\partial a_{ij}}{\partial x_i} +\frac{\partial a_{ij}}{\partial u}u_{x_i} +a=F(t,x,u,u(g_r),u_{x_k}). \]

It is clear that \(|a_{ij}|_{\alpha,Q}\le M_1,\ |b|_0\le M_2\). Therefore the estimate \(|u_{x_i}|_{\alpha,Q}\) follows from Theorem 1 of [4].

After all the estimates needed by us have been obtained, the existence of a solution of problem (1), (2) is proved by the method of continuation with respect to a parameter (see, for example, [2]); here one uses the Leray–Schauder theorem [5] on a fixed point and Friedman’s theorem [6] on the solvability of the first boundary value problem for linear parabolic equations.

Let us prove the uniqueness of the solution obtained. Suppose that problems (1), (2) have two solutions \(u(t,x)\) and \(v(t,x)\). Then the difference \(w(t,x)=u(t,x)-v(t,x)\) satisfies the equation

\[ w_t=a_{ij}(t,x,u)w_{x_i x_j} +\bigl[a_{ij}(t,x,u)-a_{ij}(t,x,v)\bigr]v_{x_i x_j} + \]

\[ +F(t,x,u,u(g_r),u_{x_k})-F(t,x,v,v(g_r),v_{x_k}). \]

We write the increment \(a_{ij}\) in the form

\[ \Delta a_{ij}=a_{ij}(t,x,u)-a_{ij}(t,x,v)= \]

\[ =w\int_0^1 \frac{\partial}{\partial u}a_{ij}\,[t,x,\tau u+(1-\tau)v]\,d\tau =B_{ij}w. \]

Similarly the increment of \(F\) is written as

\[ \Delta F=A_i w_{x_i}+H_l w(g_l)+Bw, \]

where, as is easy to see, \(H_l\ge0\). This gives for \(w\) the relation

\[ w_t=a_{ij}w_{x_i x_j}+A_i w_{x_i}+H_l w(g_l)+Hw, \]

\[ H=B_{ij}v_{x_i x_j}+B. \]

Make the substitution \(w=\eta e^{\gamma t}\). Then \(\eta\) satisfies the equation

\[ \eta_t=a_{ij}\eta_{x_i x_j}+A_i\eta_{x_i} +H_l e^{-\gamma(t-g_l(t))}\eta(g_l)+(H-\gamma)\eta . \tag{11} \]

Choose \(\gamma\) so that \(\gamma>\sum_{l=1}^q H_l+|H|\). The functions \(1/n\), \(-1/n\), for any \(n\), give respectively a positive and a negative residual when they are substituted into equation (11). Therefore, by the lemma, \(|\eta|<1/n\). Passing here to the limit as \(n\to\infty\), we obtain \(|\eta|\le0\).

Consequently \(\eta\equiv0\), and hence \(w\equiv0\). The theorem is proved.

In the case of one spatial variable, for an equation of the more general form

\[ L[u]\equiv u_t-\frac{\partial}{\partial x}a_1(t,x,u,u_x)-a(t,x,u,u(g_r),u_x)=0 \]

the following holds.

Theorem 2. Let the functions \(a_1(t,x,u,p)\) and \(a(t,x,u,u(g_r),p)\) satisfy the conditions:

1)
\[ \frac{\partial}{\partial p}a_1(t,x,u,p)\geq v(|u|)>0 \]
for \((t,x)\in Q=[0\leq x\leq x_0]\times[0,T]\) and arbitrary \(u\) and \(p\);

2)
\[ \left[\frac{\partial}{\partial x}a_1(t,x,u,0)+a(t,x,u,u(g_r),0)\right]_u\leq N;\quad 0\leq a_{u(g_l)}(t,x,0,u(g_r),0)\leq N_l; \]
\[ \left|\frac{\partial}{\partial x}a_1(t,x,0,0)+a(t,x,0,0,0)\right|\leq N_0 \]
for \((t,x)\in Q\) and arbitrary \(u\) and \(u(g_r)\);

3) for \((t,x)\in Q\), all \(p\), \(|u|\leq e^{kT}=M\), \(|u(g_r)|\leq M\), where
\[ k>|N|+\sum_{r=1}^{a}N_r+N_0, \]
the inequality
\[ |a|+\left|\frac{\partial a_1}{\partial p}\right|(1+p)^2+ \left(\,|a_1|+\left|\frac{\partial a_1}{\partial u}\right|+\left|\frac{\partial a_1}{\partial x}\right|\,\right)(1+p) \leq \mu(1+p)^2 \]
is fulfilled,

and the functions \(a_1,\dfrac{\partial a_1}{\partial p},\dfrac{\partial a_1}{\partial u},\dfrac{\partial a_1}{\partial x},a\) are continuous and satisfy a Hölder condition with respect to \(t,x,u,u(g_r),p\) with exponents \(\beta/2,\beta,\beta,\beta,\beta\), respectively;

4) the functions \(g_r(t)\), for \(t\in[0,T]\), satisfy a Hölder condition with exponent \(\beta\);

5) at the points \((0,0)\), \((0,x_0)\) the necessary compatibility conditions of the initial and boundary data are fulfilled. Then the boundary-value problem \(L[u]=0,\ u|_{x=0}=0,\ u|_{x=x_0}=0,\ u|_{t<0}=0\) has a solution belonging to the class \(C^{\beta^2,\beta^2/2}_{2,1}(\overline Q)\). This solution is unique if the following additional condition is fulfilled:

6) in every finite part of the domain
\[ \{(t,x)\in Q,\ |u|\leq M,\ |u(g_r)|\leq M,\ -\infty<p<\infty\} \]
the functions \(\dfrac{\partial a_1}{\partial x},\dfrac{\partial a_1}{\partial u},\dfrac{\partial a_1}{\partial p},a\) are differentiable with respect to \(u,u(g_r),p\).

The proof of Theorem 2 is carried out in the same way as the proof of Theorem 1, with the difference that the quantity \(|u_x|_{\alpha,Q}\) is estimated in the same way as in [1].

Remark. Let us note that in the case of the strict inequalities \(g_r(t)<t\), the conditions \(c_r(t,x)\geq0\) in the lemma and \(a_{u(g_r)}\geq0\) in Theorems 1 and 2 are not necessary.

For the boundary-value problem

\[ \left. \begin{aligned} u_t&=a_{ij}\bigl(t,x,u(g_r)\bigr)u_{x_i x_j} +a\bigl(t,x,u(g_r),u_{x_k}\bigr),\\ u|_S&=0,\quad u|_{t<0}=0 \end{aligned} \right\} \tag{12} \]

there holds

Theorem 3. Suppose that:

1) for \((t,x)\in \overline Q\) and arbitrary \(u(g_r)\) the inequalities
\[ a_{ij}\bigl(t,x,u(g_r)\bigr)\xi_i\xi_j\geq0, \]

\[ u a\bigl(t,x,u(g_r),0\bigr) \leq b u^2+\sum_{r=1}^{q} b_r u^2(g_r)+b_0, \]

\(b\geq 0,\ b_r\geq 0,\ b_0\geq 0\) are some constants;

2) for \((t,x)\in Q\), arbitrary \(p_k\) and

\[ |u(g_r)|\leq e^{\gamma T}\sqrt{\frac{b_0}{\gamma-b-\sum_{r=1}^{q} b_r}}=M,\qquad \gamma>b+\sum_{r=1}^{q} b_r, \]

the functions \(a_{ij}(t,x,u(g_r))\) and \(a(t,x,u(g_r),p_k)\) are continuous in all arguments, differentiable with respect to \(x_k\), \(u(g_r)\), \(p_k\), and satisfy the inequalities

\[ \nu \sum_{i=1}^{n}\xi_i^2 \leq a_{ij}\bigl(t,x,u(g_r)\bigr)\xi_i\xi_j \leq \mu \sum_{i=1}^{n}\xi_i^2, \]

\[ \sum_{k=1}^{n}\left|\frac{\partial a}{\partial p_k}\right|(1+p) +\sum_{r=1}^{q}\left|\frac{\partial a}{\partial u(g_r)}\right| +\sum_{k=1}^{n}\left|\frac{\partial a}{\partial x_k}\right| +|a|\leq \mu(1+p)^2, \]

\[ \sum_{ij=1}^{n}\sum_{r=1}^{q} \left|\frac{\partial a_{ij}}{\partial u(g_r)}\right|\leq \varepsilon, \]

\[ \sum_{r=1}^{q}\left|\frac{\partial a}{\partial u(g_r)}\right| \leq (1+p)^2\bigl(\varepsilon+\Phi(p)\bigr), \]

where \(\varepsilon\) is a sufficiently small number determined only by the numbers \(M,\mu,\nu,\alpha\), and \(\Phi(p)\) is such that \(\Phi(p)\to 0\) as \(p\to\infty\);

3) in every finite part of the domain
\[ \{(t,x)\in \overline Q,\ |u(g_r)|\leq M,\ -\infty<p_k<\infty\} \]
the functions \(a_{ij},a\) and their first partial derivatives with respect to \(x_k\), \(u(g_r)\), and \(p_k\) satisfy a Hölder condition in \(t,x,u(g_r),p_k\) with exponents \(\beta/2,\beta,\beta,\beta\), respectively;

4) the functions \(a_{ij}(t,x,0)\) for \((t,x)\in S\) satisfy a Lipschitz condition in \(t\);

5) the functions \(g_r(t)\) for \(t\in[0,T]\) satisfy a Hölder condition with exponent \(\beta\);

6) \(a(0,x,0,0)\equiv 0\) for \(x\in\sigma\);

7) \(\Omega\in A_2^\beta\).

Under these conditions there exists a unique solution of problem (12) of the class
\[ C_{2,1}^{\beta,\beta/2}(\overline Q). \]

To prove this theorem it is enough to obtain a priori estimates of the norms \(|u|_0\), \(\max_{\overline Q}|\nabla u|\), \(|u_{x_i}|_{\alpha,Q}\). The estimate of \(|u|_0\) can be obtained analogously to how this was done, for example, in [3] (p. 339), and the estimate of \(\max_{\overline Q}|\nabla u|\)—as in [7]. Next, the estimate of \(|u_{x_i}|_{\alpha,Q}\) can be obtained on the basis of Lemma 6 of [8] and Theorem 1 of [4].

References

1. O. A. Ladyzhenskaya, N. N. Ural’tseva. Izv. Akad. Nauk SSSR, Ser. Mat., 26, 5–52, 1962.

2. O. A. Ladyzhenskaya, N. N. Ural’tseva. Izv. Akad. Nauk SSSR, Ser. Mat., 26, 753–780, 1962.

3. O. A. Ladyzhenskaya, N. N. Ural’tseva. Linear and Quasilinear Equations of Elliptic Type. “Nauka,” Moscow, 1964.

4. A. Friedman. Journ. Math. and Mech., 9, No. 4, 539–556, 1960.

5. J. Leray, J. Schauder. UMN, 1, issue 3–4, 71–95, 1946.

6. A. Friedman. Journ. Math. and Mech., 7, No. 5, 771–791, 1958.

7. O. A. Ladyzhenskaya, N. N. Ural’tseva. UMN, 16, issue 1 (97), 19–90, 1961.

8. O. A. Oleinik, S. N. Kruzhkov. UMN, 16, issue 5 (101), 115–155, 1961.

Received by the editors
February 28, 1966

Udmurt State
Pedagogical Institute