MATHEMATICS

S. I. KHUDYAEV

THE FIRST BOUNDARY VALUE PROBLEM FOR NONLINEAR PARABOLIC EQUATIONS

(Presented by Academician I. G. Petrovskii, 13 X 1962)

1. In this paper the general nonlinear parabolic equation of second order is considered

\[ u_t - F(x,t,u,u_{x_k},u_{x_k x_l})=0 \qquad (k,l=1,2,\ldots,n). \tag{1} \]

In the cylindrical domain \(Q_{t_0T}=\{\overline{G}\times [t_0T]\}\), \(x\in G,\ t\in [t_0T]\), where \(G\) is a bounded domain of \(n\)-dimensional space with boundary \(S\), the first boundary value problem is posed:

\[ u\big|_{t=t_0}=u_0(x);\qquad u\big|_{\Gamma}=\varphi(x,t),\quad \text{where } \Gamma=\{S\times [t_0T]\}. \tag{2} \]

The function \(F(x,t,u,p_k,p_{kl})\) is assumed to be defined for \((x,t)\in Q_{t_0T}\) and for arbitrary values of the remaining arguments.

The purpose of the present paper is to establish a theorem on the local solvability of problem (1)—(2) under only certain smoothness conditions imposed on the function \(F(x,t,u,p_k,p_{kl})\), the boundary \(S\), and the boundary functions \(u_0(x)\) and \(\varphi(x,t)\). The exact formulation of these conditions will be given below. It should be noted that no conditions on the growth of the function \(F(x,t,u,p_k,p_{kl})\) with respect to the arguments \(u,p_k,p_{kl}\) \((k,l=1,\ldots,n)\) are imposed.

An analogous theorem for quasilinear equations was established by P. E. Sobolevskii \((^1)\) with the aid of the theory of fractional powers of operators and the Schauder principle.

The solution of problem (1)—(2) is constructed in the class \(C^{2+\alpha}\) by the method of successive approximations, analogous to Picard’s method for ordinary equations. This construction essentially relies on Friedman’s results \((^3)\) on an a priori estimate in the closed domain of the solution of the first boundary value problem and on the solvability of such a problem for linear parabolic equations. The uniqueness theorem for problem (1)—(2) was established in \((^2)\).

Let us make several remarks concerning notation. \(|u|^{st}_{k+\alpha}\) denotes the norm in the space \(C^{k+\alpha}(Q_{st})\), where \(Q_{st}=\{\overline{G}\times [s,t]\}\); \(k=0,1,2;\ 0\leq \alpha<1\).

The spaces \(C^{2+\alpha}(G)\) and \(C^{2+\alpha}(Q_{tt})\) are distinguished. The latter denotes the space of limit values of functions depending on \(t\), and the norms in these spaces are related by the relation

\[ |u|^{tt}_{2+\alpha}=|u|^{G}_{2+\alpha}+|u_t|^{G}_{\alpha}. \]

The class of functions \(u(x,t)\) from \(C^{2+\alpha}(Q_{t_0\tau})\) satisfying conditions (2) for \(t_0\leq t\leq \tau\), if, of course, conditions (2) allow this, is denoted by \(C^{t_0\tau}\).

1. We formulate the main result of the paper.

Theorem. Let the following conditions be satisfied:

1) In any domain

\[ R_a=\left\{(x,t)\in Q_{t_0T},\ |u|+\sum |p_i|+\sum |p_{ij}|\leq a\right\} \]

the function \(F(x,t,u,p_k,p_{kl})\) has bounded third derivatives with respect to \(u,p_k,p_{kl}\) \((k,l=1,2,\ldots,n)\), and \(F_t\) has bounded second derivatives with respect to the same arguments. Moreover, \(F,F_t\), and all the derivatives listed satisfy a Hölder condition in \((x,t)\) with exponent \(\alpha,\ 0<\alpha<1\).

2) The parabolicity condition for equation (1) in the domains \(R_a\) is fulfilled in the form

\[ \sum_{ij} F_{p_{ij}}\xi_i\xi_j \geq \nu(a)\sum_i \xi_i^2, \]

where \(\nu(a)\) is a positive nonincreasing function.

3) \(S \in A^{2+\alpha}\) (see (3)) and there exist functions \(\psi_1 \in C^{2+\alpha}(Q_{t_0T})\) and \(\psi_2 \in C^{2+\alpha}(Q_{t_0T})\) such that

\[ \psi_1(x,t_0)=u_0(x);\qquad \psi_1(x,t)\big|_{\Gamma}=\varphi(x,t); \]

\[ \psi_2(x,t_0)=w_0(x)=F(x,t_0,u_0(x),u_{0x_i},u_{0x_i x_j});\qquad \psi_2(x,t)\big|_{\Gamma}=\varphi_t(x,t). \]

Moreover, the following compatibility condition is satisfied:

\[ \varphi_{tt}(x,t_0)=\sum_{ij}F_{p_{ij}}(x,t_0,u_0,u_{0x_k},u_{0x_kx_l})\,w_{0x_i x_j}+ \]

\[ +\sum_i F_{p_i}(\cdot)w_{0x_i}+F_u(\cdot)w_0+F_t(\cdot) \]

for \(x \in S\); \((\cdot)\) denotes the same arguments as for \(F_{p_{ij}}\).

Under these conditions there exists a \(t'>t_0\) (\(t'\leq T\)) such that problem (1)—(2) has in the domain \(Q_{t_0t'}\) a unique solution \(u(x,t)\) such that \(u\in C^{2+\alpha}(Q_{t_0t'})\), \(u_t\in C^{2+\alpha}(Q_{t_0t'})\).

To prove the theorem just formulated, we consider the auxiliary boundary-value problem for the linear equation:

\[ w_t-\sum_{ij}F_{p_{ij}}(x,t,u,u_{x_k},u_{x_kx_l})w_{x_i x_j} -\sum_i F_{p_i}(\cdot)w_{x_i}-F_u(\cdot)w-F_t(\cdot); \tag{3} \]

\[ w\big|_{t=t_0}=w_0(x);\qquad w\big|_{\Gamma}=\varphi_t(x,t). \tag{4} \]

Here \(u(x,t)\) is an arbitrary function of the class \(C^{t_0\tau}\) (\(\tau\leq T\)). We note that if \(u(x,t)\) were a sufficiently smooth solution of problem (1)—(2), then \(w=u_t\) would be a solution of problem (3)—(4), and conversely: if, for some function \(u(x,t)\in C^{t_0\tau}\), \(w=u_t\) is a solution of problem (3)—(4), then \(u(x,t)\) is a solution of problem (1)—(2).

The proof of the theorem consists in constructing such a \(u\in C^{t_0t'}\) that \(u_t\) satisfies (3) and (4).

On the basis of the results of Friedman’s work (3), problem (3)—(4) is solvable for any function \(u\in C^{t_0\tau}\), and the solution \(w\in C^{2+\alpha}(Q_{t_0T})\). Thus problem (3)—(4) defines an operator \(w=Mu\), taking \(C^{t_0\tau}\) into \(C^{2+\alpha}(Q_{t_0\tau})\). We shall construct another operator:

\[ v=Nu=u_0(x)+\int_{t_0}^{t} w(x,s)\,ds. \tag{5} \]

Obviously, this operator takes functions of the class \(C^{t_0\tau}\) again into \(C^{t_0\tau}\). Lemmas 1—4 are devoted to the construction of a fixed point of the transformation (5), which will complete the proof of the theorem.

For what follows, denote by \(K_0(a)\) a common upper bound for the moduli of the function \(F\) and of all its derivatives required in condition 1) of the theorem, and also for their Hölder constants with respect to \((x,t)\) in the domain \(R_a\).

Lemma 1. Let \(u,u_1,u_2\in C^{t_0\tau}\), \(|u|^{t_0\tau}_{2+\alpha}\leq \lambda\), \(|u_i|^{t_0\tau}_{2+\alpha}\leq \lambda\), \(\tau\leq T\), \(i=1,2\). Then the following estimates hold:

\[ |Mu|^{s\tau}_{2+\alpha}\leq K_2(\lambda),\qquad t_0\leq s\leq \tau; \tag{6} \]

\[ |Mu_1-Mu_2|^{t_0\tau}_{2+\alpha}\leq K_3(\lambda)|u_1-u_2|^{t_0\tau}_{2+\alpha}. \tag{7} \]

The functions \(K_2(\lambda)\), \(K_3(\lambda)\) do not depend on \(s,\tau\) and are determined by the domain \(Q_{t_0T}\), the conditions (4), and the functions \(\nu(\lambda)\), \(K_0(\lambda)\).

Proof. For the coefficients of equation (3) it is not difficult to obtain an estimate of the norms in \(C^\alpha(Q_{s\tau})\)

\[ \left|\,\right|_{\alpha}^{s\tau}\leq K_0(\lambda)(2+\lambda)=K_1(\lambda). \tag{8} \]

The same estimate is, obviously, also valid for all second derivatives of the functions \(F\) required by the theorem. Then, according to \((^3)\), taking (8) into account, we obtain the estimate

\[ |w|_{2+\alpha}^{s\tau}\leq K\bigl(K_1(\lambda),\nu(\lambda)\bigr)\bigl(K_1(\lambda)+|\psi|_{2+\alpha}^{s\tau}\bigr). \tag{9} \]

Here \(\psi\) is any extension into the domain \(Q_{s\tau}\) of the functions \(w(x,s)\) and \(\varphi_t(x,t)\) \((s\leq t\leq \tau)\); \(K(x,y)\) is determined by the domain \(Q_{t_0T}\); the function \(\psi\) can be chosen so that \(|\psi|_{2+\alpha}^{s\tau}\leq |w|_{2+\alpha}^{t_0T}\), and for the latter one can write an estimate analogous to (9), where instead of \(|\psi|_{2+\alpha}^{s\tau}\) one may put \(|\psi_2|_{2+\alpha}^{t_0T}\) (see condition 3) of the theorem). Substituting this estimate for \(|w|_{2+\alpha}^{t_0T}\) in place of \(|\psi|_{2+\alpha}^{sT}\) in (9), we obtain (6).

If, using Hadamard’s device, we write the equation satisfied by the function \(Mu_1-Mu_2\), then inequality (9) for this equation gives the estimate (7).

Lemma 2. The following estimates are valid:

\[ |Nu|_{2+\alpha}^{t_0\tau}\leq |Nu|_{2+\alpha}^{t_0t_0} +\left[1+(\tau-t_0)^{-\alpha/2}\right]\int_{t_0}^{\tau}|Mu|_{2+\alpha}^{t_0s}\,ds; \tag{10} \]

\[ |Nu_1-Nu_2|_{2+\alpha}^{t_0\tau}\leq \left[1+(\tau-t_0)^{-\alpha/2}\right]\int_{t_0}^{\tau}|Mu_1-Mu_2|_{2+\alpha}^{t_0s}\,ds. \tag{11} \]

Proof. Consider functions \(g\in C^{2+\alpha}(Q_{t_0\tau})\) and

\[ f(x,t)=f(x,t_0)+\int_{t_0}^{t} g(x,s)\,ds. \tag{12} \]

Since the derivatives of the function \(f\) are also represented through the corresponding derivatives of the function \(g\) in the form (12), it is obvious that:

\[ |f|_{2}^{t_0\tau}\leq |f|_{2}^{t_0t_0}+\int_{t_0}^{\tau}|g|_{2}^{t_0s}\,ds. \tag{13} \]

For the Hölder constant \(H_\alpha^{t_0\tau}(f)\), from the representation (12) it is not difficult to obtain the estimate

\[ H_\alpha^{t_0\tau}(f)\leq H_\alpha^{G}(f(x,t_0)) +\int_{t_0}^{\tau}H_\alpha^{t_0s}(g)\,ds +(\tau-t_0)^{-\alpha/2}\int_{t_0}^{\tau}|g|_{0}^{t_0s}\,ds. \]

This estimate is also valid for the derivatives of \(f\); therefore, taking (13) into account and using the trivial estimate \(|g|_{2}^{t_0s}\leq |g|_{2+\alpha}^{t_0s}\), we obtain:

\[ |f|_{2+\alpha}^{t_0s}\leq |f|_{2+\alpha}^{t_0t_0} +\left[1+(\tau-t_0)^{-\alpha/2}\right]\int_{t_0}^{\tau}|g|_{2+\alpha}^{t_0s}\,ds. \tag{14} \]

By virtue of (5), estimate (14) holds for \(Nu\) and \(Nu_1-Nu_2\), which gives us (10) and (11). Note that \(|Nu|_{2+\alpha}^{t_0t_0}=\lambda_0\) does not depend on the particular choice of \(u\), but is determined only by \(u_0(x)\) and \(w_0(x)\).

Lemma 3. Let \(\lambda>\lambda_0\) and \(t_1>t_0\) be such that

\[ (t_1-t_0)^{1-\alpha/2}+(t_1-t_0)\leqslant \frac{\lambda-\lambda_0}{K_2(\lambda)} . \tag{15} \]

Then in any class \(C^{t_0\tau}\) \((\tau\leqslant t_1)\) the operator \(N\) maps the set
\(\{u: |u|_{2+\alpha}^{t_0\tau}\leqslant \lambda\}\) into itself.

Proof follows trivially from (10), (6), and (15). Take
\(t'=\min\{t_1,t_0+1\}\) and an arbitrary function \(v_0\in C^{t_0t'}\) such that
\(|v_0|_{2+\alpha}^{t_0t'}\leqslant \lambda\)*, and construct the sequence
\(v_{n+1}=Nv_n\) \((n=0,1,\ldots)\).

Lemma 4. The sequence \(v_n(x,t)\) converges to some function \(u(x,t)\) in the space \(C^{2+\alpha}(Q_{t_0t'})\), and \(Nu=u\).

Proof. Denote \(Mv_n=w_n\). Then

\[ v_{n+1}(x,t)=u_0(x,t)+\int_{t_0}^{t} w_n(x,s)\,ds . \tag{16} \]

According to (11), (7), and Lemma 3, we have

\[ |v_{n+1}-v_n|_{2+\alpha}^{t_0t} \leqslant K_3(\lambda)\,[1+(t-t_0)^{-\alpha/2}] \int_{t_0}^{t}|v_n-v_{n-1}|_{2+\alpha}^{t_0s}\,ds, \qquad t\leqslant t' . \]

Hence, by induction, we easily obtain

\[ |v_{n+1}-v_n|_{2+\alpha}^{t_0t} \leqslant 2\lambda\, \frac{\left[2K_3(\lambda)(t-t_0)^{1-\alpha/2}\right]^n}{n!}. \]

This ensures the convergence of the sequence \(v_n\). Let
\(u(x,t)=\lim\limits_{n\to\infty}v_n(x,t)\). From (7) there also follows the convergence of the sequence \(w_n\). Let
\(w(x,t)=\lim\limits_{n\to\infty}w_n(x,t)\). Since one may pass to the limit in equation (3), we have \(w=Mu\). Passing to the limit in (16), we obtain \(u=Nu\). It remains to note that \(u\) and \(w=u_t\) belong to \(C^{2+\alpha}(Q_{t_0t'})\). Lemma 4, and thereby the theorem, are proved.

1. In conclusion we note a simple consequence concerning the continuation of the solution \(u(x,t)\).

Corollary. For every \(\lambda>\lambda_0\) there exists \(s>t_0\) such that the solution \(u(x,t)\) of problem (1)—(2) exists in the domain \(Q_{t_0s}\) and, together with \(u_t\), belongs to the space \(C^{2+\alpha}(Q_{t_0s})\); moreover either \(s=T\), \(|u|_{2+\alpha}^{t_0T}\leqslant \lambda\), or \(s<T\), \(|u|_{2+\alpha}^{t_0s}=\lambda\).

Indeed, if \(t'<T\) and \(|u|_{2+\alpha}^{t_0t'}=\lambda'<\lambda\), then, taking \(u'(x)=u(x,t')\) as the initial condition, we fall under the conditions of the theorem and, consequently, the solution \(u(x,t)\) is continued into some domain \(Q_{t't''}\), where
\((t''-t')^{1-\alpha/2}+(t''-t')\leqslant (\lambda-\lambda')/K_2(\lambda)\) and
\(t''-t'\leqslant 1\). If we also observe that, by Lemma 1, the function \(K_2(\lambda)\) does not depend on \(t'\), then our assertion becomes obvious.

Institute of Chemical Physics
Academy of Sciences of the USSR

Received
11 X 1962

References

1. P. E. Sobolevskii, Tr. Mosk. matem. obshch., 10, 297 (1961).
2. L. Nirenberg, Comm. Pure and Appl. Math., 6, 2, 167 (1953).
3. A. Friedman, J. Math. and Mech., 7, 5, 771 (1958).

* As \(v_0\) one may take
\[ v_0=u_0+\int_{t_0}^{t}\psi_2(x,s)\,ds . \]
Since \(|\psi_2|_{2+\alpha}^{t_0T}\leqslant K_2(\lambda)\) for any \(\lambda>\lambda_0\) (by the definition of \(K_2(\lambda)\)), it follows from (14) and (15) that
\[ |v_0|_{2+\alpha}^{t_0t'}\leqslant \lambda . \]