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A numerical verification method for solutions of nonlinear parabolic problems. (English) Zbl 1211.35153

Summary: By using the finite element approximation and constructive a priori error estimates, a new formulation for proving the existence of solutions for nonlinear parabolic problems is presented. We present a method to estimate the norm of the linearized inverse operator for concerned nonlinear problem. Then we formulate a verification principle for solutions by using the Newton-type operator incorporating with Schauder’s fixed point theorem.

MSC:

35K58 Semilinear parabolic equations
35K20 Initial-boundary value problems for second-order parabolic equations
35A35 Theoretical approximation in context of PDEs
35A01 Existence problems for PDEs: global existence, local existence, non-existence
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
35B45 A priori estimates in context of PDEs
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