MATHEMATICS

V. A. SOLONNIKOV

ON BOUNDARY-VALUE PROBLEMS FOR GENERAL LINEAR PARABOLIC SYSTEMS

(Presented by Academician V. I. Smirnov on 6 February 1964)

In the cylinder \(Q=\Omega \times [0,T]\), where \(\Omega\) is a bounded \(n\)-dimensional domain with smooth boundary \(S\), consider the system of equations

\[ \sum_{j=1}^{m} l_{kj}\left(x,t,\frac{\partial}{\partial x},\frac{\partial}{\partial t}\right)u_j=f_k(x,t) \]

\[ (k=1,\ldots,m;\ x=(x_1,\ldots,x_n)\in\Omega,\ t\in[0,T]), \tag{1} \]

where \(l_{kj}\) are linear differential operators with complex coefficients that are smooth functions of \(x\) and \(t\).

It is assumed that there exist integers \(s_k,t_k\) \((k=1,\ldots,m,\ \max_k s_k=0)\) and an integer \(b>0\) such that the following conditions are satisfied:

1. The degree of the polynomial \(l_{kj}(x,t,i\xi\lambda,p\lambda^{2b})\) in the variable \(\lambda\) at each point \((x,t)\in Q\) does not exceed \(s_k+t_j\); moreover, if \(s_k+t_j<0\), then \(l_{kj}=0\) (here \(i=\sqrt{-1}\) and \(\xi=(\xi_1,\ldots,\xi_n)\)).

2. Let \(l_{kj}^{0}\) be the “principal part” of the polynomial \(l_{kj}\), i.e., the sum of those terms of the polynomial \(l_{kj}\) which satisfy the condition
   \[ l_{kj}^{0}(x,t,i\xi\lambda,p\lambda^{2b}) = \lambda^{s_k+t_j} l_{kj}^{0}(x,t,i\xi,p), \]
   and let \(\mathcal L_0(x,t,i\xi,p)\) be the matrix with entries \(l_{kj}^{0}(x,t,i\xi,p)\). The polynomial
   \[ L(x,t,i\xi,p)=\det \mathcal L_0(x,t,i\xi,p) \]
   must have the following properties:
   a) \(L(x,t,0,p)=\gamma(x,t)p^{2}\), \(\gamma\ne0\), \(r>0\);
   b) the roots of the polynomial \(L(x,t,i\xi,p)\) with respect to the variable \(p\), at each point \((x,t)\in Q\) and for any real \(\xi\), satisfy the inequality

\[ \operatorname{Re} p_s < -\delta |\xi|^{2b} \qquad (\delta>0,\ s=1,\ldots,r). \]

By virtue of the homogeneity of the polynomials \(l_{kj}^{0}\),

\[ L(x,t,i\xi\lambda,p\lambda^{2b}) = \lambda^{\sum_{j=1}^{m}(s_j+t_j)} L(x,t,i\xi,p). \]

From property a) it follows that

\[ \sum_{j=1}^{m}(s_j+t_j)=2br. \]

Systems for which all these conditions are satisfied will be called parabolic. Systems parabolic in the sense of I. G. Petrovskii \((^{1})\) and in the sense of T. Shirota \((^{2})\) are special cases of the general parabolic systems defined above. There exist general parabolic systems that do not belong to these classes.

We shall consider in the cylinder \(Q\) the problem consisting in finding a solution of system (1) satisfying the initial conditions

\[ \sum_{j=1}^{m} C_{\alpha j}\left(x,\frac{\partial}{\partial t}\right)u_j\bigg|_{t=0} = \varphi_{\alpha}(x) \qquad (\alpha=1,\ldots,r), \tag{2} \]

and also the boundary conditions on the lateral surface \(\Gamma=S\times[0,T]\) of the cylinder \(Q\):

\[ \sum_{j=1}^{m} B_{qj}\left(x,t,\frac{\partial}{\partial x},\frac{\partial}{\partial t}\right)u_j\bigg|_{\Gamma} =\Phi_q(x,t)\qquad (q=1,\ldots,br). \tag{3} \]

Here \(C_{\alpha j}\) and \(B_{qj}\) are linear differential operators with smooth coefficients, defined respectively in \(\Omega\) and on \(\Gamma\).

Suppose that there exist integers \(\rho_\alpha\) \((\alpha=1,\ldots,r)\) such that

\[ C_{\alpha j}(x,p)= \begin{cases} C_{\alpha j}(x)\,p^{\frac{1}{2b}(\rho_\alpha+t_j)},& \text{if }\dfrac{1}{2b}(\rho_\alpha+t_j)\text{ is a nonnegative integer},\\[6pt] 0,& \text{in all other cases}. \end{cases} \]

The matrix \(C(x,p)\) of size \(r\times m\), with elements \(C_{\alpha j}(x,p)\), must satisfy the following algebraic condition: the rows of the matrix
\[ D(x,p)=C(x,p)\widehat{\mathscr L}_0(x,0,0,p), \]
where \(\widehat{\mathscr L}_0\) is the adjugate matrix of \(\mathscr L_0\) \((\widehat{\mathscr L}_0=L\mathscr L_0^{-1})\), are linearly independent modulo the polynomial \(p^r\).

The possibility of constructing a matrix \(C\) possessing the prescribed properties for any given parabolic system is determined by the topological structure of the domain \(\Omega\) and by the matrix \(\mathscr L_0(x,0,0,1)\). If such a construction is possible, then the matrix \(C\) is determined uniquely up to certain algebraic transformations.

We now formulate the requirements imposed on the operators \(B_{qj}\). Suppose that there exist integers \(\sigma_q\) such that the degree of
\[ B_{qj}(x,t,i\xi\lambda,p\lambda^{2b}) \]
with respect to \(\lambda\) does not exceed \(\sigma_q+t_j\), and if \(\sigma_q+t_j<0\), then \(B_{qj}=0\). Let \(B^0_{qj}\) be the sum of those terms of the polynomial \(B_{qj}(x,t,i\xi,p)\) for which

\[ B^0_{qj}(x,t,i\xi\lambda,p\lambda^{2b}) = \lambda^{\sigma_q+t_j}B^0_{qj}(x,t,i\xi,p). \]

Let \(x\in S\); \(\nu(x)=(\nu_1,\ldots,\nu_n)\) be the unit vector of the inward normal to \(S\); and let \(\zeta(x)\) be a vector lying in the tangent plane to \(S\) at the point \(x\). It follows from the parabolicity condition that at each point \((x,t)\in\Gamma\), for any \(\zeta\) and \(p\) satisfying the condition

\[ \operatorname{Re}p>-\delta_1|\zeta|^{2b}, \tag{4} \]

where \(0<\delta_1<\delta\), and \(\delta\) enters into the formulation of the parabolicity condition, the polynomial \(L(x,t,i(\zeta+\tau\nu),p)\), as a function of \(\tau\), has \(br\) roots \(\tau_s^+\) with positive imaginary part and \(br\) with negative imaginary part. Let

\[ M^+(x,t,\zeta,p,\tau)=\prod_{s=1}^{br}\bigl(\tau-\tau_s^+(x,t,\zeta,p)\bigr). \]

The matrix \(B_0\) of size \(br\times m\), with elements \(B^0_{qj}(x,t,i\xi,p)\), must satisfy the following algebraic complementarity condition: at each point \((x,t)\in\Gamma\), for any tangent \(\zeta(x)\) and any \(p\) satisfying condition (4), the rows of the matrix

\[ A(x,t,i(\zeta+\tau\nu),p) = B_0(x,t,i(\zeta+\tau\nu),p)\widehat{\mathscr L}_0(x,t,i(\zeta+\tau\nu),p) \]

are linearly independent modulo the polynomial \(M^+\).

In addition, we impose one more algebraic requirement, which is probably connected with our approach to this problem. We assume that there exists a matrix \(\widetilde B_0(x,t,i\xi,p)\) of size \(br\times m\) possessing the following properties:

1) The elements \(\widetilde B^0_{qj}(x,t,i\xi,p)\) are polynomials in \(\xi_j\) and \(p\), and
\[ \widetilde B^0_{qj}(x,t,i\xi\lambda,p\lambda^{2b}) =\lambda^{\mu_q+s_j}\widetilde B^0_{qj}(x,t,i\xi,p), \]
where the \(\mu_q\) are integers.

2) The matrix \(\widetilde B_0\) satisfies the complementing condition with the transposed matrix \(\mathcal L_0^*\). If \(n=1\), then we may omit this requirement.

Let \(l>0\) be a noninteger, \(C^l(\Omega)\) the class of functions defined and \([l]\) times continuously differentiable in \(\Omega\), whose derivatives of order \([l]\) satisfy the Hölder condition with exponent \(l-[l]\), and \(C^{\,l,\frac1{2b}l}_{x,t}(Q)\) the class of functions defined in \(Q\), whose derivatives of order \([l]\) with respect to the variables \(x\) satisfy the Hölder condition in these same variables with exponent \(l-[l]\), and, moreover, whose derivatives with respect to \(t\) of order \(\left[\frac{l}{2b}\right]\) satisfy the Hölder condition in \(t\) with exponent \(\frac{l}{2b}-\left[\frac{l}{2b}\right]\). The class \(C^{\,l,\frac{l}{2b}}_{x,t}(\Gamma)\) is defined analogously. Norms in these classes are introduced in the usual way.

Under the conditions listed above, the following theorem holds.

Theorem. Let \(l\) be a noninteger satisfying the condition \(l>\max(0,\sigma_1,\ldots,\sigma_{br})\). If the coefficients of the operators \(l_{kj}\) belong to the classes \(C^{\,l-s_k,\frac1{2b}(l-s_k)}_{x,t}(Q)\), the coefficients of the operators \(C_{\alpha j}\) to the classes \(C^{\,l-\rho_\alpha}(\Omega)\), those of the operators \(B_{qj}\) to the classes \(C^{\,l-\sigma_q,\frac1{2b}(l-\sigma_q)}_{x,t}(\Gamma)\), the surface \(S\) is a Lyapunov surface of class \(C^{\,l+t_{\max}}\), \(f_j\in C^{\,l-s_j,\frac1{2b}(l-s_j)}_{x,t}(Q)\), \(\varphi_\alpha\in C^{\,l-\rho_\alpha}(\Omega)\), \(\Phi_q\in C^{\,l-\sigma_q,\frac1{2b}(l-\sigma_q)}_{x,t}(\Gamma)\), and, finally, the functions \(f_j,\varphi_\alpha,\Phi_q\) satisfy certain compatibility conditions determined by the system, the initial and the boundary conditions, then problem (1)—(3) has a unique solution \(u=(u_1,\ldots,u_m)\), with \(u_j\in C^{\,l+t_j,\frac1{2b}(l+t_j)}_{x,t}(Q)\), for which the inequality
\[ \sum_{j=1}^{m}\|u_j\|_{C^{\,l+t_j,\frac1{2b}(l+t_j)}_{x,t}(Q)} \leq C\left\{ \sum_{j=1}^{m}\|f_j\|_{C^{\,l-s_j,\frac1{2b}(l-s_j)}_{x,t}(Q)} +\right. \]
\[ \left. +\sum_{\alpha=1}^{r}\|\varphi_\alpha\|_{C^{\,l-\rho_\alpha}(\Omega)} +\sum_{q=1}^{br}\|\Phi_q\|_{C^{\,l-\sigma_q,\frac1{2b}(l-\sigma_q)}_{x,t}(\Gamma)} \right\}, \]
holds, where the constant \(C\) does not depend on \(u_j,f_j,\varphi_\alpha,\Phi_q\).

In the case where the integers \(t_j,s_j,\rho_\alpha\) are divisible by \(2b\), one can prove the existence of a solution \(u=(u_1,\ldots,u_m)\), with
\[ u_j\in W^{\,2bl+t_j,\,l+\frac{t_j}{2b}}_{px,t}(Q), \]
for which the inequality
\[ \sum_{j=1}^{m}\|u_j\|_{W^{\,2bl+t_j,\,l+\frac{t_j}{2b}}_{px,t}(Q)} \leq C'\left\{ \sum_{j=1}^{m}\|f_j\|_{W^{\,2bl-s_j,\,l-\frac{s_j}{2b}}_{px,t}(Q)} +\right. \]
\[ \left. +\sum_{\alpha=1}^{r}\|\varphi_\alpha\|_{B^{\,2bl-\rho_\alpha-\frac{2b}{p}}_{p}(\Omega)} +\sum_{q=1}^{br}\|\Phi_q\|_{B^{\,2bl-\sigma_q-\frac1p,\;\frac1{2b}\left(2bl-\sigma_q-\frac1p\right)}_{px,t}(\Gamma)} \right\} \]
holds; here \(l\) is an integer,
\[ 2bl>\max(0,\sigma_1,\ldots,\sigma_{br})+\frac1p . \]
Definitions of the norms entering this inequality may be found, for example, in (3).

The study of general boundary-value problems for systems that are parabolic in the sense of I. G. Petrovskii is the subject of works \({}^{4-9}\).

Received
29 I 1964

REFERENCES

\({}^{1}\) I. G. Petrovskii, Bull. Moscow State Univ., Section A, 1, 7 (1938).
\({}^{2}\) T. Shirota, Osaka Math. J., 9, No. 1 (1957).
\({}^{3}\) V. A. Solonnikov, DAN, 138, No. 4 (1961).
\({}^{4}\) T. Ya. Zagorskii, Ukr. Math. J., 9, No. 3 (1957).
\({}^{5}\) T. Ya. Zagorskii, Mixed problems for systems of differential equations with each derivative of parabolic type, Lvov, 1961.
\({}^{6}\) S. D. Eidelman, DAN, 142, No. 4 (1962).
\({}^{7}\) S. D. Eidelman, DAN, 149, No. 4 (1963).
\({}^{8}\) L. N. Slobodetskii, DAN, 120, No. 3 (1958).
\({}^{9}\) M. S. Agranovich, M. I. Vishik, UMN, 18, issue 1 (1963).