Geodesic
Existence theorems for solutions of parabolic equations with variable order of nonlinearity
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Differential equations and dynamical systems, Tome 270 (2010), pp. 21-32.

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We study the solvability of an initial-boundary value problem for second-order parabolic equations with variable order of nonlinearity. In the model case, the equation contains the $p$-Laplacian with a variable exponent $p(x,t)$. We prove that if the measurable exponent $p$ is separated from unity and infinity, then the problem has $W$- and $H$-solutions. 
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Yu. A. Alkhutov; V. V. Zhikov. Existence theorems for solutions of parabolic equations with variable order of nonlinearity. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Differential equations and dynamical systems, Tome 270 (2010), pp. 21-32. http://geodesic.mathdoc.fr/item/TM_2010_270_a1/

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