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Strong solutions of periodic parabolic problems with discontinuous nonlinearities

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Abstract

We study the problem of finding time-periodic solutions of a parabolic equation with the homogeneous Dirichlet boundary condition and with a discontinuous nonlinearity. We assume that the nonlinearity is equal to the difference of two superpositionally measurable functions nondecreasing with respect to the state variable. For such a problem, we prove the principle of lower and upper solutions for the existence of strong solutions without additional constraints on the “jumping-up” discontinuities in the nonlinearity. We obtain existence theorems for strong solutions of this class of problems, including theorems on the existence of two nontrivial solutions.

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Authors and Affiliations

  1. Chelyabinsk State University, Chelyabinsk, Russia

    V. N. Pavlenko

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  1. V. N. Pavlenko
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Correspondence to V. N. Pavlenko.

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Original Russian Text © V.N. Pavlenko, 2016, published in Differentsial’nye Uravneniya, 2016, Vol. 52, No. 4, pp. 528–538.

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Pavlenko, V.N. Strong solutions of periodic parabolic problems with discontinuous nonlinearities. Diff Equat 52, 505–516 (2016). https://doi.org/10.1134/S0012266116040108

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