Geodesic
Extinction of Solutions of Parabolic Equations with Variable Anisotropic Nonlinearities
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Differential equations and dynamical systems, Tome 261 (2008), pp. 16-25

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We study the Dirichlet problem for a class of nonlinear parabolic equations with nonstandard anisotropic growth conditions that generalize the evolutional $p(x,t)$-Laplacian. We study the property of extinction of solutions in finite time. In particular, we show that the extinction may take place even in the borderline case when the equation becomes linear as $t\to\infty$. 
@article{TM_2008_261_a1,
     author = {S. N. Antontsev and S. I. Shmarev},
     title = {Extinction of {Solutions} of {Parabolic} {Equations} with {Variable} {Anisotropic} {Nonlinearities}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {16--25},
     publisher = {mathdoc},
     volume = {261},
     year = {2008},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TM_2008_261_a1/}
}
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S. N. Antontsev; S. I. Shmarev. Extinction of Solutions of Parabolic Equations with Variable Anisotropic Nonlinearities. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Differential equations and dynamical systems, Tome 261 (2008), pp. 16-25. http://geodesic.mathdoc.fr/item/TM_2008_261_a1/