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Time-space estimates of solutions to general semilinear parabolic equations. (English) Zbl 1106.35027

This paper is concerned with estimates for solutions of the nonlinear parabolic equation \[ u_ t + \sum_{| \alpha| \leq 2m} a_ \alpha (x)\partial^ \alpha u = F(u, \partial u, \dots, \partial ^{2m-1}u), \] with either a Cauchy initial condition or a Cauchy-Dirichlet initial-boundary condition. The functions \(a_ \alpha\) are assumed sufficiently smooth and elliptic (that is, the form \[ (-1)^ m \text{Re} \sum_{| \alpha| \leq 2m} a_ \alpha (x)\xi^ \alpha \] is uniformly positive on the sphere \(| \xi| =1\)). The conditions on the nonlinear function \(F\) are too complicated to relate exactly here, but they include both sums of terms of the form \(C_ \beta \partial^\beta f_ \beta(u)\) with appropriate growth of the Lipschitz constants for the functions \(f_ \beta\) and also (for the Cauchy problem) sums of terms of the form \(C_ j (- \Delta)^{j/2}f_ j(u)\). Various parameters determine the appropriate space in which the solution is estimated but the spaces are the general form \(L^ q((0,T);L^ p(\Omega))\) with the exponents \(p\) and \(q\) related in a suitable fashion to (a) the growth of the \(f_ \beta\) (or \(f_ j\)) and (b) the exponent of integrability of the initial data. From the estimates, the author deduces various existence results, especially when the initial data are small.

MSC:

35K55 Nonlinear parabolic equations
35B45 A priori estimates in context of PDEs
35D05 Existence of generalized solutions of PDE (MSC2000)
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