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Nonlinear parabolic equations with nonstandard growth. (English) Zbl 1349.35176
Summary: In this paper, we study the solvability of the initial-boundary value problem for second-order nonlinear parabolic equations with nonstandard growth conditions and \(L^2\)-source terms. In the model case, these equations include the \(p\)-Laplacian with a variable exponent \(p(x,t)\). We prove that if the variable exponent \(p\) is bounded away from both 1 and \(+\infty\) and is \(\log\)-Hölder continuous, then the problem has a weak solution which satisfies the energy equality.

MSC:
35K55 Nonlinear parabolic equations
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
35D30 Weak solutions to PDEs
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