UDC 517.946.8:517.946.9

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

M. V. Kozlova, V. V. Podgornov

A system of equations of order \(2b\), parabolic in the sense of Petrovskii, is considered:
\[ \frac{\partial u(t,x)}{\partial t} = \sum_{l=2b} A_l\bigl(t,x,D^k u(g_i(t),x)\bigr)D^l u(t,x) + F\bigl(t,x,D^k u(g_i(t),x)\bigr). \tag{1} \]

Here
\[ x=(x_1,\ldots,x_n);\qquad u(t,x)=(u_1(t,x),\ldots,u_N(t,x)); \]
\[ D^k=\frac{\partial^{k_1+\cdots+k_n}}{\partial x_1^{k_1}\cdots \partial x_n^{k_n}}; \qquad k=k_1+\cdots+k_n\geq 0,\quad k\leq 2b-1; \]
\[ g_i(t)\leq t,\qquad i=1,\ldots,r; \]
\(A_l(t,x,\ldots,y_j,\ldots)\) are square matrices of order \(N\);
\(F(t,x,\ldots,y_j,\ldots)=(F_1,\ldots,F_N)\), \(|y_j|<M\), \(j=0,\ldots,rs\), where \(s\) is the number of all possible derivatives \(D^k\).

In the cylinder \(Q=\Omega\times[0,T]\) one seeks a solution of system (1) satisfying the boundary conditions:
\[ u(t,x)\big|_{t\leq 0}=0, \]
\[ \left.\frac{\partial^\nu u(t,x)}{\partial n^\nu}\right|_S=0, \qquad \nu=0,\ldots,b-1, \tag{2} \]
where \(\Omega\subset R_n\) is a domain bounded by a convex surface of Lyapunov type; \(S\) is the lateral surface of the cylinder; \(n\) is the direction of the inner normal to \(S\).

We shall denote by \(C^{(\alpha,\beta,\ldots,1,\ldots)}(P)\) the space of vector functions \(f(t,x,\ldots,y_j,\ldots)\) that are continuous and bounded in the domain \(P\) and satisfy a Hölder condition in \(t,x,\ldots,y_j,\ldots\), respectively with exponents \(\alpha,\beta,\ldots,1,\ldots\), uniformly in \(P\) \((0<\alpha<1,\;0<\beta<1)\). By \(P_M\) we denote the domain \(\{t,\ x\in Q,\ |y_j|<M\}\).

In \(Q\) define the class \(B(Q,2b,M,M_1,\alpha)\) of functions \(\{v(t,x)\}\), continuously differentiable with respect to \(x\) up to order \(2b-1\), bounded together with their derivatives by the constant \(M\), and satisfying, together with derivatives up to order \(2b-1\), a Hölder condition in \(x\) and \(t\) with constant \(M_1\) and exponents \(\alpha\) and \(\dfrac{\alpha}{2b}\), respectively.

Theorem 1. Problem (1), (2) has in \(Q'\{x\in\Omega,\ t\in[0,\Delta],\ \Delta>0\}\) a unique solution, continuous together with the derivatives entering the system, satisfying, together with the derivatives, a Hölder condition in \(x\) and \(t\) with exponents \(\gamma\) and \(\dfrac{\gamma}{2b}\), respectively \((\gamma<\alpha)\), if the following conditions are fulfilled:

1) the elements of the matrix \(A_l(t,x,\ldots,y_j,\ldots)\in C^{\left(\frac{\alpha}{2b},\,\alpha,\ldots,1,\ldots\right)}(P_M)\);

2) the vector-function \(F(t,x,\ldots,y_j,\ldots)\in C^{(0,\alpha,\ldots,1,\ldots)}(P_M)\);

3) the functions \(g_i(t)\in C^{(1)}[0,T]\).

Proof. Define in \(B(Q',2b,M,M_1,\alpha)\) the operator

\[ w=V(v)=\int_0^t d\tau \int_\Omega G_v(t,x,\tau,\xi)F(\tau,\xi,D^k v(g_i(\tau),\xi))\,d\xi, \tag{3} \]

where \(G_v(t,x,\tau,\xi)\) is the Green matrix of the problem

\[ \frac{\partial \overline{w}(t,x)}{\partial t} = \sum_{l=2b} A_l(t,x,D^k v(g_i(t),x))D^l\overline{w}(t,x),\quad k\leq 2b-1, \]

\[ \left.\frac{\partial^\nu \overline{w}(t,x)}{\partial n^\nu}\right|_S\equiv 0,\quad \overline{w}(0,x)\equiv 0 \quad(\text{see }[1]). \]

We shall show that, for sufficiently small \(\Delta>0\), the operator \(V\) maps the class of functions \(B(Q',2b,M,M_1,\alpha)\) into itself. First note that for the derivatives of the Green matrix the following estimates hold:

\[ |D^kG(t,x,\tau,\xi)| \leq C(t-\tau)^{-\frac{n+k}{2b}}\times \]

\[ \times \exp\left( -\mu\frac{|x-\xi|^{\frac{2b}{2b-1}}}{(t-\tau)^{\frac{1}{2b-1}}} \right),\quad k\leq 2b, \tag{4} \]

where \(C,\mu\) are constants depending on the upper bounds of the moduli of the coefficients of the system, their Hölder constants, the numbers \(a,T\), and the parabolicity condition of the system.

If \(v\in B(Q',2b,M,M_1,\alpha)\), then, using the conditions of the theorem, as well as the estimates (4), one can show that the following inequalities hold:

\[ |D^k w(t,x)|\leq C_v t^{\frac{2b-k+\alpha}{2b}},\quad 0<k\leq 2b, \]

\[ |w(t,x)|\leq C_v t, \]

\[ |\Delta D^k w(t,x)| = |D^k w(t,x)-D^k w(t',x')| \leq \tag{5} \]

\[ \leq C_v d^\gamma t^{\frac{2b-k+\alpha-\gamma}{2b}},\quad 0<k\leq 2b, \]

\[ |\Delta w(t,x)|\leq C_v d^\alpha t^{\frac{2b-\alpha}{2b}} . \]

for points \((t,x)\), \((t',x') \in Q\), \(t \leq t'\); here
\[ d=\sqrt{|x-x'|^2+(t'-t)^{1/b}}; \]
\(\gamma=\alpha\) for \(k<2b\); \(\gamma\) is any positive number smaller than \(\alpha\) for \(k=2b\); the constant \(C_v\) depends on the upper bounds for the moduli of the coefficients of system (1) and \(F\), on their Hölder constants, on the numbers \(T,\alpha,\gamma,2b\), and on the parabolicity condition of system (1) (cf. [2], p. 109).

Let \(t \in [0,\Delta]\). Choosing \(\Delta\) so that
\[ C_v \Delta^{\frac{2b-k+\alpha}{2b}}<M \quad\text{and}\quad C_v \Delta^{\frac{2b-k+\alpha-\gamma}{2b}}\leq M_1,\quad \Delta<1, \]
from (5) we obtain
\[ |D^k w(t,x)|<M, \]
\[ |\Delta D^k w(t,x)|\leq M_1 d^\gamma,\qquad k\leq 2b, \tag{6} \]
i.e.
\[ w\in B(Q',2b,M,M_1,\alpha). \]

Consider the equation
\[ v=V(v). \tag{7} \]
Set
\[ v_{-1}\equiv 0,\qquad v_m=V(v_{m-1}),\quad m\geq 0, \]
i.e.
\[ v_m(t,x)=\int_0^t d\tau \int_\Omega G_{v_{m-1}}(t,x,\tau,\xi)\, F\bigl(\tau,\xi,D^k v_{m-1}(g_i(\tau),\xi)\bigr)\,d\xi. \tag{8} \]

We shall establish the convergence of the sequence \(\{v_m\}\).

Introduce the notation
\[ \varepsilon_m(t)=\sup_{x\in\Omega}\sum_{k<2b} \left|D^k\bigl(v_m(t,x)-v_{m-1}(t,x)\bigr)\right|, \qquad m=0,1,\ldots \]

Since the operator \(V\) maps the set of functions \(\{v_m\}\) into itself, system (8) is equivalent to the system of differential equations
\[ \frac{\partial v_m(t,x)}{\partial t} = \sum_{l=2b} A_l\bigl(t,x,D^k v_{m-1}(g_i(t),x)\bigr)D^l v_m(t,x) + \]
\[ + F\bigl(t,x,D^k v_{m-1}(g_i(t),x)\bigr). \]

Write the equation for the difference \(v_{m+1}-v_m\):
\[ \frac{\partial}{\partial t}[v_{m+1}-v_m] = \sum_{l=2b} A_l\bigl(t,x,D^k v_m(g_i(t),x)\bigr)D^l[v_{m+1}-v_m]+ \]
\[ + \left[ \sum_{l=2b} A_l\bigl(t,x,D^k v_m(g_i(t),x)\bigr) - \sum_{l=2b} A_l\bigl(t,x,D^k v_{m-1}(g_i(t),x)\bigr) \right]D^l v_m+ \]
\[ + F\bigl(t,x,D^k v_m(g_i(t),x)\bigr) - F\bigl(t,x,D^k v_{m-1}(g_i(t),x)\bigr) \equiv \]
\[ \equiv \sum_{l=2b} A_l\bigl(t,x,D^k v_m(g_i(t),x)\bigr)D^l[v_{m+1}-v_m]+\Phi(t,x), \]

whence

\[ v_{m+1}-v_m=\int_0^t d\tau \int_\Omega G_{v_m}(t,x,\tau,\xi)\Phi(\tau,\xi)d\xi . \tag{9} \]

Using the conditions of the theorem and inequalities (6), we estimate \(\Phi(t,x)\):

\[ \begin{aligned} |\Phi(t,x)| \leq {}& \left|\sum_{k=2b}\left[A_l\left(t,x,D^k v_m(g_i(t),x)\right)-\right.\right.\\ &\left.\left.-A_l\left(t,x,D^k v_{m-1}(g_i(t),x)\right)\right]\right| \,|D^l v_m(t,x)|+\\ &+\left|F\left(t,x,D^k v_m(g_i(t),x)\right)- F\left(t,x,D^k v_{m-1}(g_i(t),x)\right)\right| \leq\\ \leq {}& C_1 \sum_{k<2b}\left|D^k \sum_{i=1}^r \left(v_m(g_i(t),x)-v_{m-1}(g_i(t),x)\right)\right| \leq C_1 \sum_{i=1}^r \varepsilon_m(g_i(t)). \end{aligned} \tag{10} \]

Apply the operator \(D^k,\ k \leq 2b-1\), to both sides of system (9):

\[ D^k(v_{m+1}-v_m)=\int_0^t d\tau \int_\Omega D^kG_{v_m}(t,x,\tau,\xi)\Phi(\tau,\xi)d\xi . \]

Using estimates (4) and (10), we obtain

\[ \begin{aligned} |D^k(v_{m+1}-v_m)| \leq {}& C_2\int_0^t \sum_{i=1}^r \varepsilon_m(g_i(\tau))(t-\tau)^{-\frac{n+k}{2b}} \times\\ &\times \int_\Omega \exp\left(-\mu \frac{|x-\xi|^{\frac{2b}{2b-1}}}{(t-\tau)^{\frac{1}{2b-1}}} \right)d\xi \leq\\ \leq {}& C_2\int_0^t (t-\tau)^{-\frac{2b-1}{2b}} \sum_{i=1}^r \varepsilon_m(g_i(\tau))d\tau \end{aligned} \]

and, consequently,

\[ \varepsilon_{m+1}(t)\leq C_3\int_0^t (t-\tau)^{-\frac{2b-1}{2b}} \sum_{i=1}^r \varepsilon_m(g_i(\tau))d\tau,\quad m=0,1,\ldots \tag{11} \]

Denote

\[ z_k(t)=\sum_{m=0}^k \varepsilon_m(t). \tag{12} \]

Then

\[ z_{k+1}(t)\leq \varepsilon_0(t)+C_3\int_0^t (t-\tau)^{-\frac{2b-1}{2b}} \sum_{i=1}^r z_k(g_i(\tau))d\tau . \]

Consider a sequence \(\overline z_k(t)\) such that

\[ \bar z_{k+1}(t)=C_3\int_0^t (t-\tau)^{-\frac{2b-1}{2b}}\sum_{i=1}^r \bar z_k(g_i(\tau))\,d\tau+\varepsilon_0(t), \]

\[ \bar z_0(t)=z_0(t)=\varepsilon_0(t). \]

Obviously, the sequence \(\bar z_k(t)\) converges uniformly to \(z(t)\), where \(z(t)\) is the continuous solution of the equation

\[ z(t)=C_3\int_0^t (t-\tau)^{-\frac{2b-1}{2b}}\sum_{i=1}^r z(g_i(\tau))\,d\tau+\varepsilon_0, \]

\[ z(t)\big|_{t\leq 0}=0. \]

Let us show that \(z_k\leq \bar z_k\). Indeed, since \(\bar z_0=z_0\), we have

\[ z_1(t)\leq C_3\int_0^t (t-\tau)^{-\frac{2b-1}{2b}}\sum_{i=1}^r z_0(g_i(\tau))\,d\tau+\varepsilon_0(t)= \]

\[ = C_3\int_0^t (t-\tau)^{-\frac{2b-1}{2b}}\sum_{i=1}^r \bar z_0(g_i(\tau))\,d\tau+\varepsilon_0(t)=\bar z_1(t). \]

Suppose that \(z_k(t)\leq \bar z_k(t)\). Then

\[ z_{k+1}(t)\leq C_3\int_0^t (t-\tau)^{-\frac{2b-1}{2b}}\sum_{i=1}^r z_k(g_i(\tau))\,d\tau+\varepsilon_0(t)\leq \]

\[ \leq C_3\int_0^t (t-\tau)^{-\frac{2b-1}{2b}}\sum_{i=1}^r \bar z_k(g_i(\tau))\,d\tau+\varepsilon_0(t)=\bar z_{k+1}(t). \]

Thus, the sequence \(z_k(t)\) is majorized by the uniformly convergent sequence of functions \(\bar z_k(t)\), whence, from (12), the uniform convergence of the series \(\sum_{m=0}^{\infty}\varepsilon_m(t)\) follows. Hence we conclude the uniform convergence of the sequence \(D^k v_m(t,x)\), \(k\leq 2b-1\).

Let \(u(t,x)=\lim_{m\to\infty} v_m(t,x)\). Since the sequence \(\{D^k v_m(t,x)\}\) converges uniformly, \(D^k u(t,x)=\lim_{m\to\infty} D^k v_m(t,x)\).

Let us show that \(u(t,x)\) is a solution of equation (7). Indeed, by virtue of inequality (11) we have

\[ \sup_{x\in\Omega}\left|D^k(u-V(u))\right| \leq \sup_{x\in\Omega}\left|D^k(u-v_{m+1})\right|+ \]

\[ +\sup_{x\in\Omega}\left|D^k\bigl(V(v_m)-V(u)\bigr)\right|\leq \]

\[ \leq \sup_{x\in\Omega}\left|D^k(u-v_{m+1})\right| + C_3\int_0^t (t-\tau)^{-\frac{2b-1}{2b}} \sum_{i=1}^r \sup_{x\in\Omega}\left|D^k\bigl(u(g_i)-v_m(g_i)\bigr)\right|\,d\tau \to 0. \]

as \(m \to \infty\). Thus, \(u(t,x)\) satisfies the equation

\[ u(t,x)=\int_0^t d\tau \int_\Omega G_u(t,x,\tau,\xi)\, F\bigl(\tau,\xi,D^k u(g_i(\tau),\xi)\bigr)\,d\xi . \]

And since \(u(t,x)\in B(Q',2b,M,M_1,\alpha)\) and satisfies the boundary conditions (2), \(u(t,x)\) is the desired solution of problem (1), (2).

Let us prove the uniqueness of the solution obtained. Suppose that problem (1), (2) has two solutions \(u_1(t,x)\) and \(u_2(t,x)\). Then the difference \(\eta(t,x)=u_1(t,x)-u_2(t,x)\) satisfies the system

\[ \frac{\partial \eta}{\partial t} = \sum_{l=2b} A_l\bigl(t,x,D^k u_2(g_i(t),x)\bigr)D^l\eta(t,x) +\overline{\Phi}(t,x), \]

\[ \left.\frac{\partial^\nu \eta}{\partial n^\nu}\right|_S=0, \qquad \left.\eta\right|_{t\le 0}\equiv 0, \]

where

\[ \overline{\Phi}(t,x)= \sum_{l=2b}\bigl[ A_l\bigl(t,x,D^k u_1(g_i(t),x)\bigr) - \]

\[ - A_l\bigl(t,x,D^k u_2(g_i(t),x)\bigr) \bigr]D^l u_1 + \]

\[ + F\bigl(t,x,D^k u_1(g_i(t),x)\bigr) - F\bigl(t,x,D^k u_2(g_i(t),x)\bigr), \]

whence

\[ \eta(t,x)=\int_0^t d\tau \int_\Omega G_{u_2}(t,x,\tau,\xi)\Phi(\tau,\xi)\,d\xi . \tag{13} \]

Analogously to inequality (10), we find

\[ |\overline{\Phi}(t,x)|\le C_4\sum_{i=1}^r \overline{\varepsilon}\bigl(g_i(t)\bigr), \tag{14} \]

where

\[ \overline{\varepsilon}(t)= \sup_{x\in\Omega}\sum_{k<2b}|D^k\eta(t,x)|. \]

From (13), using estimates (4) and (14), we obtain

\[ \overline{\varepsilon}(t)\le C_5\int_0^t (t-\tau)^{-\frac{2b-1}{2b}} \sum_{i=1}^r \overline{\varepsilon}\bigl(g_i(\tau)\bigr)\,d\tau . \]

Since the equation

\[ y(t)=C_5\int_0^t (t-\tau)^{-\frac{2b-1}{2b}} \sum_{i=1}^r y\bigl(g_i(\tau)\bigr)\,d\tau, \]

\[ \left.y(t)\right|_{t\le 0}\equiv 0 \]

has the unique zero solution, by virtue of the theorem on integral inequalities [3] we obtain \(\overline{\varepsilon}(t)\le 0\), whence \(\eta(t,x)\equiv 0\).

The theorem is proved.

Remark. If one assumes \(M=\infty\), then for the parabolic system

\[ \frac{\partial u(t,x)}{\partial t} = \sum_{k=2b} A^{(k_1,\ldots,k_n)}(t,x)D^{2b}u(t,x) + F\bigl(t,x,D^k u(g_i(t),x)\bigr), \tag{15} \]

where \(A^{(k_1,\ldots,k_n)}(t,x)\) are square matrices of order \(N\), the following nonlocal theorem, analogous to Theorem 1, can be proved.

Theorem 2. In the domain \(Q\) there exists a unique solution of problem (15), (2), continuous together with the derivatives entering the system, and satisfying, together with the derivatives, a Hölder condition in \(x\) and \(t\) with exponents \(\gamma\) and \(\dfrac{\gamma}{2b}\), respectively, if the following conditions are fulfilled:

1) the elements of the matrices \(A^{(k_1,\ldots,k_n)}(t,x)\in C^{\left(0,\frac{\alpha}{2b},\alpha\right)}(Q)\);

2) the vector-functions \(F(t,x,\ldots,y_j,\ldots)\in C^{(0,\alpha,\ldots,1,\ldots)}(P_\infty)\);

3) the functions \(g_i(t)\in C^{(1)}\).

Analogous theorems can also be proved for the Cauchy problem.

The authors thank the participants of the Izhevsk mathematical seminar for their attention to the work.

References

1. V. P. Mikhailov. DAN SSSR, 132, No. 2, 1960.
2. S. D. Eidelman. Parabolic Systems. Moscow, Nauka, 1964.
3. Z. B. Tsalyuk. DAN, 134, No. 1, 1960.