UDC 517.946

MATHEMATICS

S. I. POHOZHAEV

ON THE SOLVABILITY OF QUASILINEAR PARABOLIC SYSTEMS

(Presented by Academician L. S. Pontryagin on 16 XII 1969)

This note contains an application of the theorem on the solvability of a nonlinear equation in a Banach space from paper (¹) to boundary-value problems for quasilinear parabolic (in the sense of Douglis—Nirenberg—Solonnikov (²)) systems of differential equations. For simplicity in formulating the theorem we have restricted ourselves to the case when all right-hand sides of the system belong to the space \(L_p(Q)\), and to the case of linear boundary conditions.

It turns out that, for the solvability of the boundary-value problems considered below for quasilinear parabolic systems, it is sufficient that the corresponding a priori estimate for the solutions of these problems exist. Thus, in this case there is no need, after obtaining the a priori estimate, to carry out an additional proof of the existence of a solution.

In the note we make essential use of the results of V. A. Solonnikov (³) on boundary-value problems for linear parabolic systems.

1. Definitions. Let \(\Omega\) be a bounded domain in the space \(R^n\) with boundary \(S\), \(Q=\Omega\times[0,T]\), \(0<T<+\infty\), and \(\Gamma=S\times[0,T]\). Let
\(u(x,t)=(u_1(x,t),\ldots,u_m(x,t))\) be a real vector-function. Let
\(\alpha=(\alpha_0,\alpha_1,\ldots,\alpha_n)\) be an integer nonnegative multi-index,
\(|\alpha|=\alpha_0+\alpha_1+\cdots+\alpha_n\),
\(D^\alpha u_k=\partial^{|\alpha|}u_k(x,t)/\partial t^{\alpha_0}\partial x_1^{\alpha_1}\cdots\partial x_n^{\alpha_n}\).
Put
\(|\alpha|_b=2b\alpha_0+\alpha_1+\cdots+\alpha_n\), where \(b\) is some positive integer.

Consider a quasilinear system of equations of the form

\[ \sum_{j=1}^{m}\ \sum_{|\gamma|_b=t_j} a_{ij}^{\gamma}(x,t,D^{\alpha^k}u_k)\,D^\gamma u_j + a_i(x,t,D^{\alpha^k}u_k) = f_i(x,t). \tag{1} \]

Here \(i,k=1,\ldots,m\); \(t_j=2bt'_j\), where \(t'_j\) \((j=1,\ldots,m)\) are nonnegative integers and
\(\sum_{j=1}^{m} t_j=2br\) (\(r\) is a positive integer); \(\gamma\) and \(\alpha^k\) are integer nonnegative multi-indices with
\(|\alpha^k|_b\le t_k-1\); \(a_{ij}^{\gamma}\), \(a_i\), and \(f_i\) are real functions. In addition, the functions \(a_{ij}^{\gamma}\) and \(a_i\) contain neither \(D^{\alpha^k}u_k\) nor \(u_k\) if \(t_k=0\).

To the system (1) we assign boundary conditions of the form

\[ \sum_{j=1}^{m}\ \sum_{|\gamma|_b\le \sigma_s+t_j} b_{sj}^{\gamma}(x,t)\,D^\gamma u_j\big|_{\Gamma} = \Phi_s(x',t). \tag{2} \]

Here \(s=1,\ldots,br\); \(\sigma_s\) are integers with \(\max_s \sigma_s<0\); \(\gamma\) is an integer nonnegative multi-index; \(b_{sj}^{\gamma}\) and \(\Phi_s\) \((x'\in S)\) are real functions. In addition, \(b_{sj}^{\gamma}\equiv0\) if \(\sigma_s+t_j<0\).

We consider the system (1) under the boundary conditions (2) and the initial zero conditions

\[ \partial^i u_j/\partial t^i\big|_{t=0}=0,\quad i=0,\ldots,t'_j-1;\quad j=1,\ldots,m. \tag{3} \]

At the same time, if \(t_j' = 0\), then no initial condition is imposed for \(u_j\).
With the system (1) with conditions (1) and (3) we associate the real spaces

\[ H_1(Q)=\prod_{j=1}^{m}\dot W_p^{\,t_j,t_j'}(Q),\qquad H_2(Q)=\prod_{i=1}^{m}L_p(Q) \]

and

\[ H_3(\Gamma)=\prod_{s=1}^{br}\dot W_p^{-\sigma_s-\frac1p,\;\frac1{2b}\left(-\sigma_s-\frac1p\right)}(\Gamma), \]

with \(p>n+2b\) and with norms

\[ \|u\|_{H_1(Q)}=\sum_{j=1}^{m}\|u_j\|_{p,Q}^{(t_j)},\qquad \|f\|_{H_2(Q)}=\sum_{i=1}^{m}\|f_i\|_{p,Q} \quad\text{and}\quad \|\Phi\|_{H_3(\Gamma)}=\sum_{s=1}^{br}\|\Phi_s\|_{p,\Gamma}^{(-\sigma_s-1/p)}. \]

The spaces \(\dot W_p^{2bl,l}(Q)\) with integer \(l\ge 0\) and \(\dot W_p^{k,\frac1{2b}k}(\Gamma)\) with noninteger \(k>0\) are defined in the work of V. A. Solonnikov \({}^{3}\).

2. Solvability theorem. Problem (1)—(2)—(3) is considered under the following assumptions:

Condition I (smoothness condition). Let the real functions
\(a_{ij}^{\gamma}(x,t,D^{\alpha_k}u_k)\) and
\(a_i(x,t,D^{\alpha_k}u_k)\) be continuous and have continuous first derivatives with respect to the variables corresponding to \(D^{\alpha_k}u_k\), for \((x,t)\in \overline Q\) and for arbitrary real values of the variables corresponding to \(D^{\alpha_k}u_k\). Let the real coefficients \(b_{sj}^{i\gamma}\) of the boundary operators belong to the class

\[ C_{x,t}^{-\sigma_s-\frac1p+\varepsilon,\;\frac1{2b}\left(-\sigma_s-\frac1p\right)}(\Gamma) \]

with \(\varepsilon>0\). Let the boundary \(S\) belong to the class \(C^{t_{\max}}\) \((t_{\max}=\max_j t_j)\).

Condition II (parabolicity condition). Let the linear (with respect to \(v\)) system

\[ \sum_{j=1}^{m}\sum_{|\gamma|_b=t_j} a_{ij}^{\gamma}(x,t,D^{\alpha_k}u_k)D^{\gamma}v_j=\psi_i(x,t) \qquad (i=1,\ldots,m) \tag{4} \]

for every fixed vector-function \(u(x,t)\) from \(H_1(Q)\) be parabolic in the sense of work \({}^{2}\) (with \(s_i=0\)) (see also \({}^{4}\), Ch. VII, §§ 8, 9).

Condition III (complementarity condition). Let, for every fixed vector-function \(u(x,t)\) from \(H_1(Q)\), the boundary conditions (2) satisfy the complementarity condition in the sense of work \({}^{2}\) (see also \({}^{4}\), Ch. VII, §§ 8, 9) with respect to the linear (in \(v\)) system (4).

Condition IV (condition for the existence of an a priori estimate). Let, for every possible solution \(u(x,t)\in H_1(Q)\) of problem (1), (2), (3) with
\(\|f\|_{H_2(Q)}+\|\Phi\|_{H_3(\Gamma)}\le K\), the inequality
\(\|u\|_{H_1(Q)}\le C(K)\) hold, where \(C(K)<+\infty\) for \(K<+\infty\).

Taking into account the definitions, we formulate the solvability theorem for problem (1), (2), (3).

Theorem. Let Conditions I, II, III, IV be satisfied and let \(p>n+2b\). Then for any vector-functions \(f(x,t)\in H_2(Q)\) and \(\Phi(x',t)\in H_3(\Gamma)\) there exists a solution \(u(x,t)\in H_1(Q)\) of problem (1), (2), (3).

The proof of the theorem is based on the use of Theorem 2 from work \({}^{1}\) and the results of V. A. Solonnikov \({}^{3}\) on the solvability of boundary-value problems for linear parabolic systems. We note that, in the proof of the theorem, new norms, equivalent to the original ones, are introduced in the spaces \(H_2(Q)\) and \(H_3(\Gamma)\) in order to ensure uniform convexity of the Banach space \(H_2(Q)\times H_3(\Gamma)\).

Remark 1. If, in addition, one assumes that for any fixed \(u(x,t)\) and \(w(x,t)\) from \(H_1(Q)\) the system linear with respect to \(v(x,t)\)

\[ \sum_{j=1}^{m}\sum_{|\gamma|_b=t_j}\int_{0}^{1} a_{ij}^{\gamma}(x,t,D^{\alpha_k}u_k+\tau D^{\alpha_k}w_k)\,d\tau\cdot D^{\gamma}v_j =\psi_i(x,t) \qquad (i=1,\ldots,m) \]

is parabolic in the sense of Douglis—Nirenberg—Solonnikov and the boundary conditions (2) satisfy the complementing condition (see (2); (4), Ch. VII, §§ 8, 9) with respect to this system, which is linear in \(v\), then the assertion on uniqueness in the space \(H_1(Q)\) of the solution of problem (1), (2), (3) is valid.

Remark 2. In the case when the order of the derivatives \(D^{\alpha^k}u_k\) entering into the coefficients \(a_{ij}^{\gamma}(x,t,D^{\alpha^k}u_k)\) is such that \(|\alpha^k|_b < t_k-1\), the exponent \(p\) may be decreased to an exponent \(p_1>1\) such that \(|\alpha^k|_b < t_k-(n+2b)/p_1\) for \(k=1,\ldots,m\). In this case the functions \(a_i(x,t,D^{\beta^k}u_k)\) (\(\beta^k\) are integral nonnegative multiindices with \(|\beta^k|_b \le t_k-1\)) and their derivatives with respect to the variables corresponding to \(D^{\beta^k}u_k\) must satisfy power-growth conditions with respect to \(D^{\gamma^k}u_k\) with \(|\gamma^k|_b \ge t_k-(n+2b)/p_1\) \((|\gamma^k|_b \le t_k-1)\).

Example. Let \(\Omega\) be a bounded interval \([a,b]\subset \mathbf R\). Consider in the rectangle \([a,b]\times[0,T]\), with arbitrary \(T>0\) \((T<+\infty)\), the following boundary-value problem:

\[ \frac{\partial u_i}{\partial t} -\sum_{k=1}^{m} a_{ik}(x,t,u)\frac{\partial^2 u_k}{\partial x^2} +a_i(t,u)\left(\frac{\partial u_i}{\partial x}\right)^2 +b_i(x,t,u,u_x)=f_i(x,t), \tag{5} \]

\[ u_i(a,t)=u_i(b,t)=0 \quad \text{for } t\in[0,T], \tag{6} \]

\[ u_i(x,0)=0 \quad \text{for } x\in[a,b], \tag{7} \]

where \(u=(u_1(x,t),\ldots,u_m(x,t))\), \(u_x=(\partial u_1(x,t)/\partial x,\ldots,\partial u_m(x,t)/\partial x)\), \(i=1,\ldots,m\).

Here \(a_{ik}(x,t,u)\) \((i,k=1,\ldots,m;\ u=(u_1,\ldots,u_m))\) are real continuous functions with continuous first derivatives with respect to the variables \(u_1,\ldots,u_m\) for \((x,t,u)\in[a,b]\times[0,T]\times\mathbf R^m\), and such that, for any \((x,t,u)\in[a,b]\times[0,T]\times\mathbf R^m\) and \(\xi=(\xi_1,\ldots,\xi_m)\in\mathbf R^m\), the inequality

\[ \sum_{i,k=1}^{m} a_{ik}(x,t,u)\xi_k\xi_i \ge a_0(|u|)|\xi|^2 \quad \text{with} \quad a_0(|u|)\ge c_0>0, \]

holds, where \(c_0\) is a constant, \(a_0(\rho)\) is a continuous function for \(\rho\ge0\), \(|v|^2=\sum_{i=1}^{m}v_i^2\) for \(v=(v_1,\ldots,v_m)\in\mathbf R^m\); \(a_i(t,u)\) are real continuous functions with continuous derivatives with respect to the variables \(u_1,\ldots,u_m\) for \((t,u)\in[0,T]\times\mathbf R^m\), with \(a_i(t,0)\equiv0\) when \(u_1=\cdots=u_m=0\), and such that for any \((t,u)\in[0,T]\times\mathbf R^m\) and \(\xi\in\mathbf R^m\) the inequality

\[ \sum_{i,k=1}^{m}\frac{\partial a_i(t,u)}{\partial u_k}\xi_i^3\xi_k \ge a(|u|)\sum_{i=1}^{m}\xi_i^4 \quad \text{with} \quad a(|u|)\ge0, \]

holds, where \(a(\rho)\) is a continuous function for \(\rho\ge0\); \(b_i(x,t,u,p)\) \((p=(p_1,\ldots,p_m))\) are real continuous functions with continuous derivatives with respect to the variables \(u_1,\ldots,u_m\) and \(p_1,\ldots,p_m\) for \((x,t,u,p)\in[a,b]\times[0,T]\times\mathbf R^m\times\mathbf R^m\), and such that for any \((x,t,u,p)\in[a,b]\times[0,T]\times\mathbf R^m\times\mathbf R^m\) the inequality

\[ \sum_{i=1}^{m} b_i^2(x,t,u,p) \le a_0(|u|)\left[c\left(|p|^2+|u|^2+1\right) +\left(\frac{4}{3}-\varepsilon\right)a(|u|)\sum_{i=1}^{m}p_i^4\right], \]

holds, where the constant \(c>0\) and \(\varepsilon\) is an arbitrarily small positive number.

For the boundary-value problem (5), (6), (7), with \(f_i(x,t)\in L_4(Q)\) \((i=1,\ldots,m;\ Q=[a,b]\times[0,T])\), there exists an a priori estimate for \(\|u\|_{\overset{\circ}{W}{}^{2,1}_4(Q)}\).

\[ \left(\|u\|_{\overset{\circ}{W}{}^{2,1}_{4}(Q)}^{4} = \sum_{i=1}^{m}\|u_i\|_{\overset{\circ}{W}{}^{2,1}_{4}(Q)}^{4}\right). \]

Applying the theorem and Remark 1 to this theorem, we obtain the following assertion.

If the assumptions indicated here are satisfied, then the boundary-value problem (5), (6), (7), for any fixed vector function \(f(x,t)=(f_1(x,t),\ldots,f_m(x,t))\) with \(f_i(x,t)\in L_4(Q)\) \((i=1,\ldots,m)\), has a unique solution \(u(x,t)\) with \(u_i(x,t)\in \overset{\circ}{W}{}^{2,1}_{4}(Q)\) \((i=1,\ldots,m;\ Q=[a,b]\times[0,T])\).

Remark. In connection with this example we note that a broad class of quasilinear (and nonlinear) parabolic systems with one spatial variable was considered by S. N. Kruzhkov \((^5)\).

Moscow
Power Engineering Institute

Received
11 XII 1969

REFERENCES

\(^{1}\) S. I. Pokhozhaev, Functional Analysis and Its Applications, 3, no. 2, 80 (1969).
\(^{2}\) V. A. Solonnikov, Proceedings of the V. A. Steklov Mathematical Institute, Academy of Sciences of the USSR, 83 (1965).
\(^{3}\) V. A. Solonnikov, ibid., 102, 137 (1967).
\(^{4}\) O. A. Ladyzhenskaya, V. A. Solonnikov, N. N. Ural’tseva, Linear and Quasilinear Equations of Parabolic Type, Moscow, 1967.
\(^{5}\) S. N. Kruzhkov, Doklady Akademii Nauk, 187, no. 3, 510 (1969).