S. D. Eidelman

On the Cauchy Problem for Nonlinear and Quasilinear Parabolic Systems

(Presented by Academician I. G. Petrovskii, 15.V.1957)

The present note is devoted to the exposition of theorems on the correct solvability of the Cauchy problem for nonlinear and quasilinear parabolic systems in the sense of I. G. Petrovskii \((^1)\), whose proofs are based on the application of the fundamental matrices of solutions of linear parabolic systems constructed and studied by the author in \((^2)\).

1. Let us consider the nonlinear parabolic system

\[ \frac{\partial u}{\partial t} = F\left( t,x,u,\ldots, \frac{\partial^{2b}u}{\partial x_1^{k_1}\cdots \partial x_n^{k_n}} \right); \tag{1} \]

\[ F\left( t,x,u,\ldots, \frac{\partial^{2b}u}{\partial x_1^{k_1}\cdots \partial x_n^{k_n}} \right) \]

is a one-column matrix whose elements are functions of \(t, x_1,\ldots,x_n\), the vector-function \(u\), and its derivatives up to order \(2b\) with respect to \(x_1,\ldots,x_n\).

Definition. System (1) will be called parabolic in the sense of I. G. Petrovskii in the domain

\[ \Pi_1\left\{t_0 \le t \le T;\ -\infty < x_s < \infty;\ \left| \frac{\partial^m u}{\partial x_1^{k_1}\cdots \partial x_n^{k_n}} \right| \le M,\ m=0,1,\ldots,2b\right\}, \]

if the roots of the equation

\[ \det\left\{ \left\| \sum_{\Sigma k_s=2b} \frac{\partial F_i}{ \partial\left\{ \dfrac{\partial^{2b}u_j}{\partial x_1^{k_1}\cdots \partial x_n^{k_n}} \right\}} (i\sigma_1)^{k_1}\cdots(i\sigma_n)^{k_n} \right\|_{i,j=1}^{N} -\lambda E \right\}=0 \tag{2} \]

satisfy the inequalities \(\operatorname{Re}\lambda<-\delta\) for

\[ \left( x,t,u,\ldots, \frac{\partial^{2b}}{\partial x_1^{k_1}\cdots \partial x_n^{k_n}} \right)\in \Pi_1 \]

and any real \(\sigma_1,\ldots,\sigma_n\),

\[ \sum_{s=1}^{n}\sigma_s^2=1. \]

The Cauchy problem is studied:

\[ u\big|_{t=+t_0}=\varphi(x) \tag{3} \]

for system (1).

Theorem 1. If \(F(t,x,y_1,\ldots,y_l)\) has \(r_1=2b+2\) continuous and bounded derivatives with respect to \(x_1,\ldots,x_n; y_1,\ldots,y_l\), satisfying the Lipschitz condition with respect to \(y_1,\ldots,y_l\) in the domain \(\Pi_1\), and, moreover, the continuity in \(t\) of

\[ \frac{\partial F_i}{ \partial\left[ \dfrac{\partial^{2b}u_j}{\partial x_1^{k_1}\cdots \partial x_n^{k_n}} \right]} \]

is uniform with respect to \(x_1,\ldots,x_n; y_1,\ldots,y_l\) from \(\Pi_1\); if \(\varphi(x)\) has \(r_2=4b\) continuous in the sense of Hölder and bounded derivatives with respect to \(x\), then for \(t_0<t\le t_0+\Delta\) there exists a solution of the Cauchy problem (3) for system (1), having \(r_3=4b\) continuous and bounded derivatives with respect to \(x_1,\ldots,x_n\). If, however, \(r_1=2b+3\), \(r_2=4b+1\), then \(r_3=4b+1\),

then \(r_3=4b+1\), and the solution is unique in the class of functions having \(4b\) bounded derivatives that are continuous in the Hölder sense. The solution obtained depends continuously on the initial data in the following sense: if \(u_1(x,t)\), \(u_2(x,t)\) are solutions of the Cauchy problem for system (1), constructed from the initial data \(\varphi_1(x)\) and \(\varphi_2(x)\), and
\[ \left|D_x^s(\varphi_1(x)-\varphi_2(x))\right|<\varepsilon \quad (s=0,1,\ldots,2b), \]
then
\[ \left|u_1(x,t)-u_2(x,t)\right|<M_1\varepsilon, \]
where \(M_1\) depends on \(M,T,\delta\) and on the constants bounding the norm of \(F\) and its derivatives.

If \(r_1=2b+3+p\), then every solution of system (1) that has \(4b\) bounded derivatives, continuous in the Hölder sense with respect to \(x_1,\ldots,x_n\), in the strip
\[ \Pi_1\{t_0\leq t\leq T,\ -\infty<x_s<\infty,\ s=1,2,\ldots,n\}, \]
has \(4b+1+p\) continuous derivatives with respect to \(x_1,\ldots,x_n\) for \(t>t_0\).

The proof of Theorem 1 is based on reducing the solution of the Cauchy problem (1), (3) to the solution of an equivalent Cauchy problem for a certain quasilinear system, and on solving the latter by the method of successive approximations. In doing so, it is essential that the fundamental matrices of solutions of linear parabolic systems can be constructed under very small restrictions imposed on the coefficients.

1. For quasilinear parabolic systems
   \[ \frac{\partial u}{\partial t} = P_0\left(t,x,u,\ldots,\frac{\partial^m u}{\partial x_1^{k_1}\ldots \partial x_n^{k_n}};\frac{\partial}{\partial x}\right)u + F\left(t,x,u,\ldots,\frac{\partial^m u}{\partial x_1^{k_1}\ldots \partial x_n^{k_n}}\right), \tag{4} \]
   where \(0\leq m\leq 2b-1\), \(P_0\left(t,x,y;\frac{\partial}{\partial x}\right)\) is a differential operator of order \(2b\), the following theorem is valid.

Theorem 2. Suppose that: 1) the coefficients of the operator
\[ P_0\left(t,x,y_1,\ldots,y_p;\frac{\partial}{\partial x}\right) \]
are defined in the domain
\[ \Pi_3\{t_0\leq t\leq T;\ -\infty<x_s<\infty,\ s=1,\ldots,n;\ |y_j|\leq M,\ j=1,\ldots,p\} \]
and are continuous in \(\Pi_3\) with respect to \(t\), uniformly in \(x\) and \(y\) from \(\Pi_3\); \(F(t,x,y)\) is defined and continuous with respect to \(t\) in \(\Pi_3\); 2) the coefficients of the operator \(P_0\left(t,x,y;\frac{\partial}{\partial x}\right)\) and \(F(t,x,y)\) have, in the domain \(\Pi_3\), \(r_1=m+2\) continuous and bounded derivatives with respect to \(x_1,\ldots,x_n;\ y_1,\ldots,y_p\), satisfying the Lipschitz condition with respect to \(y_1,\ldots,y_p\); the initial function \(\varphi(x)\) has \(r_2=2b+m\) bounded derivatives continuous in the Hölder sense.

Then, for \(t_0<t\leq t_0+\Delta\), the Cauchy problem (4), (3) has a solution, bounded together with all its derivatives up to order \(r_3=2b+m\); if \(r_1=m+3,\ r_2=2b+m+1\), then \(r_3=2b+m+1\), and the solution is unique in the class of functions having \(2b+m\) bounded derivatives continuous in the Hölder sense. The solution obtained depends continuously on the initial data in the sense described in Theorem 1 \((s=0,1,\ldots,m)\). If \(r_1=m+3+p\), then every solution of system (4) having \(4b+m\) bounded derivatives with respect to \(x_1,\ldots,x_n\), continuous in the Hölder sense, in the strip \(\Pi_2\), has \(2b+m+p+1\) continuous derivatives for \(t>t_0\).

1. In the case of systems close to linear,
   \[ \frac{\partial u}{\partial t} = P_0\left(t,x;\frac{\partial}{\partial x}\right)u + F\left(t,x,u,\ldots,\frac{\partial^{2b-1}u}{\partial x_1^{k_1}\ldots \partial x_n^{k_n}}\right); \tag{5} \]
   \[ u|_{t=t_0}=\varphi(x);\qquad |\varphi(x)|\leq c\exp[k|x|^q];\qquad q=\frac{2b}{2b-1}, \tag{6} \]
   a theorem is established on the correct solvability of the Cauchy problem (5), (6) in the class of rapidly growing functions.

Theorem 3*. 1) Suppose that the coefficients of
\[ P_0\left(t,x;\frac{\partial}{\partial x}\right) \]
are bounded and satisfy the Hölder condition with respect to \(x_1,\ldots,x_n\) with exponent \(\alpha\),

\[ \text{* This theorem was obtained by the author in 1955 and reported at the 3rd All-Union Mathematical Congress (2).} \]

\(0<\alpha\leqslant 1\), in the strip \(\Pi_2\), and the continuity in \(t\) is uniform with respect to \(x_1,\ldots,x_n\) from \(\Pi_2\); \(F(t,x,y_1,\ldots,y_l)\) is defined in the domain \(\Pi_4\{t_0\leqslant t\leqslant T;\ -\infty<x_s<\infty,\ s=1,\ldots,n;\ -\infty<y_i<\infty,\ i=1,\ldots,l\}\) and satisfies the conditions:

\[ |F(t,x,0)|\leqslant C_1\exp\{k|x|^q\}, \]

\[ |F(t,x,z_1,\ldots,z_l)-F(t,x,y_1,\ldots,y_l)|\leqslant f(|z-y|). \]

in every finite domain with respect to \(x_1,\ldots,x_n;\ y_1,\ldots,y_l\); \(f(h)>0\) does not depend on \(t,x,z,y\);

\[ \int_0^\eta \frac{f(h)\,dh}{h}<+\infty, \]

\[ |F(t,x,y_1,\ldots,y_l)-F(t,x,w_1,\ldots,w_l)|\leqslant B\sum_{s=1}^l |y_s-w_s|. \]

Then there exists a solution of the Cauchy problem (6) for the system (5), \(u(x,t)\), defined in the strip
\[ \Pi_5\left\{-\infty<x_s<\infty,\ s=1,2,\ldots,n;\ t_0<t\leqslant t_0+\min\left[T,\left(\frac{c}{k}\right)^{2b-1}\right]\right\} \]
(the positive constant \(c\) is determined by the coefficients of the operator \(P_0\) and by \(T\)) and satisfying the estimate
\[ |u(x,t)|\leqslant M_1\exp[k_1|x|^q], \tag{7} \]
which depends continuously on the initial data in the sense that, if
\[ \sup_x\{|\varphi_1(x)-\varphi_2(x)|\exp[-k|x|^q]\}<\delta, \]
then
\[ \sup_x\{|u_1(x,t)-u_2(x,t)|\exp[-k_1|x|^q]\}\leqslant Re\,\delta. \]

2) If, in addition, the coefficients of \(P_0\left(t,x,\dfrac{\partial}{\partial x}\right)\) have \(2b\) bounded derivatives, continuous in the sense of Hölder, with respect to \(x_1,\ldots,x_n\), then the solution obtained is unique in the class of functions
\[ \sup_x\{|u(x,t)|\exp[-k_1|x|^q]\}\leqslant C_1. \]

Let us note that in the case of linear systems Theorem 3 establishes the existence of a solution of the Cauchy problem under the assumption that the coefficients of the operator \(P_0\left(t,x;\dfrac{\partial}{\partial x}\right)\) are continuous in the sense of Hölder with respect to \(x_1,\ldots,x_n\), while from the coefficients at the lower-order derivatives it is required only that they satisfy the generalized Hölder condition.

Chernivtsi
State University

Received
9 V 1957

REFERENCES

¹ I. G. Petrovsky, Bull. Moscow State Univ., 1, no. 7 (1938). ² S. D. Eidelman, Mat. sbornik, 38 (80), 51 (1956); Proc. 3rd All-Union Math. Congress, 1, 1956.