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Coercive solvability of the nonlocal boundary value problem for parabolic differential equations. (English) Zbl 0996.35027
Summary: The nonlocal boundary value problem, $v'(t) + Av(t) = f(t)$ $(0 \leq t \leq 1)$, $v(0) = v(\lambda) + \mu$ $(0 < \lambda \leq 1)$, in an arbitrary Banach space $E$ with the strongly positive operator $A$, is considered. The coercive stability estimates in Hölder norms for the solution of this problem are proved. The exact Schauder’s estimates in Hölder norms of solutions of the boundary value problem on the range $\{0 \leq t \leq 1$, $x \in {\Bbb{R}}^n\}$ for $2m$-order multidimensional parabolic equations are obtained.

35K35Higher order parabolic equations, boundary value problems
47J35Nonlinear evolution equations
35K90Abstract parabolic equations
Full Text: DOI EuDML