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Zbl 1098.35073
Misawa, Masashi
Existence of a classical solution for linear parabolic systems of nondivergence form. (English)
[J] Commentat. Math. Univ. Carol. 45, No. 3, 475-482 (2004). ISSN 0010-2628

The author proves the existence and uniqueness of classical solutions to Cauchy-Dirichlet boundary value problem for linear parabolic systems $$ \partial _t u^i - A_{i,j}^{\alpha , \beta }D_{\alpha }D_{\beta } u^j - B_{i,j}^{\beta }D_{\beta } u^j - C_{i,j} u^j = f^i \ \text {in}\ Q = (0,T)\times \Omega ,\quad i=1,\dots ,n $$ complemented by zero initial and boundary conditions. The coefficients $A, B, C$ and right hand sides $f$ are supposed to be uniformly $\delta $ - H\" older continuous on $Q$ (with respect to the parabolic metric) and the matrix $A$ is supposed to be symmetric and satisfy the uniform strong ellipticity condition. Moreover, the right hand side $f$ is assumed to fulfill a compatibility condition $f = 0$ on $\{t = 0\}\times \partial \Omega $. Under these conditions the existence and uniqueness of of classical solution $u \in C^{1,2}(Q)$ with H\" older continuous first time derivative and second space derivatives is deduced from Schauder estimates (given in {\it W. Schlag} [Commun. Partial Differ. Equations 21, No. 7-8, 1141-1175 (1996; Zbl 0864.35023)]) by continuation method. The main ingredience of the proof is a use of energy estimates and Schauder estimates to get a bound for $L^{\infty }$ norm of a solution. \par In the last section the results are applied to gradient flows for $p$-harmonic maps between Riemannian manifolds.
[Jana Stará (Praha)]

MSC 2000:: *35K50 Systems of parabolic equations, boundary value problems
35K45 Systems of parabolic equations, initial value problems
35D10 Regularity of generalized solutions of PDE

Keywords: linear parabolic systems; nondivergence form; classical solutions; Schauder estimates; gradient flows; $p$-harmonic maps

Citations: Zbl 0864.35023

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