MATHEMATICS

L. I. KAMYNIN and V. N. MASLENNIKOVA

ON THE SOLUTION IN THE LARGE OF THE FIRST BOUNDARY-VALUE PROBLEM FOR A QUASILINEAR PARABOLIC EQUATION

(Presented by Academician S. L. Sobolev on 12 XI 1960)

In the present paper we study the first boundary-value problem for a quasilinear parabolic equation of the form

[
Lu \equiv \sum_{i,j=1}^{n} a_{ij}(x,t)\frac{\partial^2 u}{\partial x_i \partial x_j}
+ \sum_{i=1}^{n} b_i(x,t,u)\frac{\partial u}{\partial x_i}
- \frac{\partial u}{\partial t}
= f(x,t,u,\nabla u),
\tag{1}
]

where (\nabla u=(\partial u/\partial x_1,\partial u/\partial x_2,\ldots,\partial u/\partial x_n)), in noncylindrical domains (D). A. Friedman studied an analogous problem (for (b_i(x,t,u)\equiv b_i(x,t))), establishing in paper ((^1)) an a priori ((1+\delta))-estimate for the solution of the first boundary-value problem for a linear parabolic equation. However, under the restrictions imposed by A. Friedman on (f(x,t,u,\omega)), the existence of a solution was proved locally (with respect to (T)). We shall consider the question of existence and uniqueness of the solution of the first boundary-value problem for equation (1) in the large, i.e., for arbitrary (T).

Let (D) be an ((n+1))-dimensional domain of the space ((x_1,x_2,\ldots,x_n;t)\equiv(x,t)), bounded by two hyperplanes (t=0) and (t=T>0) and by a closed surface (S) lying between these hyperplanes. Let (\Omega) be the base of (D), i.e. (\Omega=\overline D\cap{t=0}). We shall call the set (\Gamma=S\cup\Omega) the normal boundary of the domain. Following A. Friedman ((^1)), we introduce the following norms:

[
|v|0^D=\sup|v(x,t)|,\qquad
|v|\alpha^D=|v|_0^D+H\alpha^D[v],
]

[
H_\alpha^D[v]=\sup_{P_1,P_2\in D}
\frac{|v(P_1)-v(P_2)|}{[d(P_1,P_2)]^\alpha},
]

where the distance between two points
(P_1(\bar x_1,\bar x_2,\ldots,\bar x_n;\bar t)) and
(P_2(\overline{\overline{x}}_1,\overline{\overline{x}}_2,\ldots,\overline{\overline{x}}_n;\overline{\overline{t}}))
is defined by

[
d(P_1,P_2)=
\left(\sum_{i=1}^{n}(\bar x_i-\overline{\overline{x}}_i)^2
+\left|\bar t-\overline{\overline{t}}\right|\right)^{1/2}.
\tag{2}
]

Further,

[
|v|{1+\alpha}^D
=
|v|\alpha^D
+
\sum_{i=1}^{n}
\left|\frac{\partial v}{\partial x_i}\right|_\alpha^D,
]

[
|v|{2+\alpha}^D
=
|v|^D
+
\sum_{i=1}^{n}
\left|\frac{\partial v}{\partial x_i}\right|{1+\alpha}^D
+
\left|\frac{\partial v}{\partial t}\right|\alpha^D.
]

I. With respect to the lateral surface (S) it is assumed that it can be covered by a finite number of spheres (W_j), and in each sphere (W_j) po-

the piece of the surface (S_j) falling into it admits, for some (i), a representation of the form

[
x_i=h(x_1,x_2,\ldots,x_{i-1},x_{i+1},\ldots,x_n;t),
]
[
(x_1,x_2,\ldots,x_{i-1},x_{i+1},\ldots,x_n,t)\in \Sigma_j,
]

where the function (h) has on (\Sigma_j) derivatives with respect to (x_k) up to the second order inclusive, satisfying the Hölder condition (with exponent (\alpha;\ 0<\alpha<1)) and a first derivative with respect to (t), also satisfying the Hölder condition (with exponent (\alpha)); here the distance between the points (P_1(\bar x,\bar t)) and (P_2(\bar{\bar x},\bar{\bar t})) in the Hölder condition is taken from (2). In addition it is assumed that (\partial h/\partial x_k) on (\Sigma_j) satisfies the Lipschitz condition with the usual distance

[
\rho(P_1,P_2)=\left(\sum_{i=1}^{n}(\bar x_i-\bar{\bar x}_i)^2+(\bar t-\bar{\bar t})^2\right)^{1/2}.
\tag{3}
]

Let the quasilinear operator (L) in (1) be parabolic for ((x,t)\in \overline D), i.e., for all real vectors ((\lambda_1,\lambda_2,\ldots,\lambda_n))

[
\sum_{i,j=1}^{n} a_{ij}(x,t)\lambda_i\lambda_j \geq a_0\sum_{i=1}^{n}\lambda_i^2 .
\tag{4}
]

Let the coefficients and the right-hand side of equation (1) satisfy the conditions:

II. For all ((x,t)\in \overline D,\ |u|<\infty,\ \partial f(x,t,u,0)/\partial u \geq b_0) ((b_0) is a constant).

III. In the domain ((x,t)\in \overline D,\ |w|<\infty) (\left(|w|^2=\sum_{i=1}^{n}w_i^2\right)) and (|u|\leq K\equiv)

[
\equiv \left(\sup_{\Gamma}|\psi|+\frac{\sup |f(x,t,0,0)|}{b_0+\gamma}\right)e^{\gamma T}
]

((\gamma>0) is a constant for which (\gamma+b_0>0)), the following conditions are fulfilled: the functions (a_{ij}(x,t), b_i(x,t,u), f(x,t,0,0), \partial f(x,t,u,0)/\partial u), and (\partial f(x,t,u,w)/\partial w_i) ((i=1,2,\ldots,n)) satisfy in ((x,t)) the Hölder condition (with exponent (\alpha;\ 0<\alpha<1)), and in (u) and (w_i) ((i=1,2,\ldots,n)) the Hölder conditions (with exponent (\beta,\ 0<\beta\leq 1)):

[
|a_{ij}(x,t)-a_{ij}(\bar x,\bar t)|\leq A_1[d(P_1,P_2)]^\alpha
\tag{5}
]

(here and below (P_1=P_1(x,t);\ P_2=P_2(\bar x,\bar t)) and (d(P_1,P_2)) is taken from (2));

[
|b_i(x,t,u)-b_i(\bar x,\bar t,\bar u)|\leq B_1[d(P_1,P_2)]^\alpha+B_2|u-\bar u|^\beta,
\tag{6}
]

[
|f(x,t,0,0)-f(\bar x,\bar t,0,0)|\leq C_1[d(P_1,P_2)]^\alpha,
]

[
\left|\frac{\partial f(x,t,u,0)}{\partial u}-\frac{\partial f(\bar x,\bar t,\bar u,0)}{\partial u}\right|
\leq C_2[d(P_1,P_2)]^\alpha+C_3|u-\bar u|^\beta;
\tag{7}
]

[
\left|\frac{\partial f(x,t,u,w)}{\partial w_i}\right|\leq C_4
\qquad (i=1,2,\ldots,n);
]

[
\left|\frac{\partial f(x,t,u,w)}{\partial w_i}-\frac{\partial f(\bar x,\bar t,\bar u,\bar w)}{\partial w_i}\right|
\leq D_1[d(P_1,P_2)]^\alpha+
]

[
+\,D_2|u-\bar u|^\beta+D_3\left[\sum_{i=1}^{n}(w_i-\bar w_i)^2\right]^{\beta/2}
\qquad (i=1,2,\ldots,n).
\tag{8}
]

IV. The functions (a_{ij}(x,t)) on (\Sigma_j) satisfy, with respect to ((x,t)), the Lipschitz condition with the usual distance (3).

V. Let there exist in the domain (\overline D) a function (\Psi(x,t)), coinciding on the normal boundary (\Gamma) of the domain (D) with the prescribed boundary function (\psi(x,t)), and such that (|\Psi|_{2+\alpha}^{D}<\infty).

Theorem 1. If the lateral surface (S), the coefficients of the quasilinear parabolic equation (1), and the boundary function (\psi(x,t)) satisfy all the conditions (4), I—V indicated above, then there exists a solution (u(x,t)), continuous in (\overline D), of equation (1) with the prescribed boundary conditions

[
u|_{\Gamma}=\psi(x,t),
\tag{9}
]

and constants (M) and (\lambda) ((0<\lambda\leq \alpha\beta<1)) can be found such that in (\overline D) the inequality

[
|u|{2+\lambda}^{D}\leq M\bigl(|f(x,t,0,0)|\bigr),}+|\Psi|_{2+\alpha
\tag{10}
]

holds, where (M) depends on the domain (D), the surface (S), and the constants (\alpha,\beta,\lambda,K,a_0,A_1,B_1,B_2,C_1,C_2,C_3,C_4,D_1,D_2,D_3).

The proof of Theorem 1 uses an a priori estimate for the modulus of a solution of the problem (1), (9), A. Friedman’s theorem on the ((1+\delta))-estimate for a solution of the first boundary-value problem for a linear parabolic equation (1), A. Friedman’s theorem on the existence and the ((2+\alpha))-estimate for a solution of the first boundary-value problem for a linear parabolic equation (2), and Schauder’s fixed-point theorem.

Theorem 2 (A. Friedman ([1])). Let (S) be any closed surface and let the quasilinear operator (L) be parabolic in (\overline D). If (b_i(x,t,u)) and (f(x,t,u,w)) are locally Lipschitz continuous with respect to (u), i.e., for (|u|\leq N)

[
|b_i(x,t,u_1)-b_i(x,t,u_2)|\leq B_2(N)|u_1-u_2|,
]

[
|f(x,t,u_1,w)-f(x,t,u_2,w)|\leq C_2(N)|u_1-u_2|,
]

then in (D) there can exist at most one solution of the first boundary-value problem (1), (9), continuous in (\overline D).

Remark. From Theorem 2, for (\beta=1) it follows that the solution of the first boundary-value problem (1), (9), whose existence was proved in Theorem 1, is unique.

Let us note one generalization of Theorem 2 on uniqueness for a more general quasilinear parabolic equation.

Theorem 3. Let (S) be any closed surface and let the quasilinear operator

[
\Lambda u=\sum_{i,j=1}^{n} a_{ij}(x,t,u,\nabla u)\frac{\partial^2 u}{\partial x_i\partial x_j}-\frac{\partial u}{\partial t}
]

be parabolic in (\overline D), i.e., for ((x,t)\in\overline D)

[
\sum_{i,j=1}^{n} a_{ij}(x,t,u,w)\lambda_i\lambda_j
\geq
a(u,w)\sum_{i=1}^{n}\lambda_i^2,
\tag{11}
]

where (a(u,w)>0) is a nonincreasing function of ((|u|+|w|)) for ((|u|+|w|)<\infty).

If (a_{ij}(x,t,u,w)) and (f(x,t,u,w)) are locally Lipschitz continuous with respect to (u), then there can exist at most one solution of the first boundary-value problem for the equation

[
\Lambda u\equiv f(x,t,u,\nabla u)
\tag{12}
]

with boundary condition (9), continuous in (\overline D) and having in (\overline D) bounded derivatives (\partial u/\partial x_i,\ \partial^2u/\partial x_i\partial x_j\ (i,j=1,2,\ldots,n)).

With the aid of Lemma 1, Theorem 4 is proved.

Lemma 1. If, for the function (f(x,t,u,w)), continuous in all arguments for (|u|<\infty), the inequality

[
|f(x,t,u,0)| \leq C_5 + C_6 |u|,
\tag{13}
]

holds, then for every solution, continuous in (\overline D), of problem (12), (9) (where (\Lambda) from (12) with continuous coefficients (a_{ij}(x,t,u,\nabla u)) satisfies (11)), the following a priori estimate holds:

[
\sup_{\overline D} |u(x,t)| \leq K_1 \equiv
\left(\sup_{\Gamma} |\psi| + \frac{C_5}{\gamma-C_6}\right)e^{\gamma T}
\tag{14}
]

((\gamma>0) is any constant for which (\gamma-C_6>0)).

Theorem 4. Let the lateral surface (S) satisfy condition I; let the boundary function (\psi(x,t)) satisfy condition V; let (a_{ij}(x,t)), continuous in the Hölder sense (5), satisfy in (\overline D) the parabolicity condition (4), as well as condition IV. Suppose that for all (|u|<\infty) (13) is fulfilled, and that in the domain ((x,t)\in \overline D), (|w|<\infty), (|u|\leq K_1) (where the constant (K_1) is taken from (14)) the functions (b_i(x,t,u)), (f(x,t,u,0)), (\partial f(x,t,u,w)/\partial w_i) are Hölder continuous, i.e. (6), (8) are fulfilled, and

[
|f(x,t,u,0)-f(\bar x,\bar t,\bar u,0)|
\leq C_7[d(P_1,P_2)]^\alpha + C_8|u-\bar u|^\beta,
]

and for (\partial f(x,t,u,w)/\partial w_i) inequality (7) holds. Then there exists a solution (u(x,t)), continuous in (\overline D), of the first boundary-value problem (1), (9), for which (10) is valid, where (M) depends on the domain (D), the surface (S), and on (\alpha,\beta,\lambda,A_1,B_1,B_2,C_4,C_5,C_6,C_7,C_8,D_1,D_2,D_3).

Remark. In A. Friedman’s paper ((^1)), for equation (1) ((b_i(x,t,u)\equiv b_i(x,t))) it is proved that: 1) if one does not require the existence of (\partial f/\partial w_i), but imposes the condition

[
|f(x,t,u,w)| \leq C_5 + C_6 |u| + C_9 |w|^\delta,
\tag{15}
]

where (0\leq \delta<1), then the solution of problem (1), (9) exists globally; 2) if in (15) (\delta=1) and one does not require the existence of (\partial f/\partial w_i), then the solution of problem (1), (9) exists for small (C_9); 3) if (f(x,t,u,w)) is locally Hölder continuous, then the solution of problem (1), (9) exists locally (with respect to (T)).

Note added in proof. The existence of a solution of problem (1), (9) was proved by the authors for a lateral surface (S) admitting a barrier at each point, and for a boundary function (\psi) continuous on (\Gamma).

Mathematical Institute named after V. A. Steklov
Academy of Sciences of the USSR

Received
11 XI 1960

REFERENCES

1. A. Friedman, J. Math. and Mech., 9, No. 4, 539 (1960).
2. A. Friedman, J. Math. and Mech., 7, No. 5, 771 (1958).