Geodesic
Existence of weak solutions to an elliptic-parabolic equation with variable order of nonlinearity
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Differential equations. Mathematical physics, Tome 139 (2017), pp. 44-58.

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We consider an equation with variable nonlinearity of the form $|u|^{p(x)}$, in which the parabolic term can vanish, i.e., in the corresponding domain the parabolic equation becomes “elliptic.” Under a weak monotonicity conditions (nonstrict inequality) we prove the existence of a solution to the first mixed problem in a cylinder with a bounded base.
Keywords: weak solution, variable nonlinearity
Mots-clés : elliptic-parabolic equation, existence of solutions. 
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     author = {F. Kh. Mukminov and \`E. R. Andriyanova},
     title = {Existence of weak solutions to an elliptic-parabolic equation with variable order of nonlinearity},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
     pages = {44--58},
     publisher = {mathdoc},
     volume = {139},
     year = {2017},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/INTO_2017_139_a4/}
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F. Kh. Mukminov; È. R. Andriyanova. Existence of weak solutions to an elliptic-parabolic equation with variable order of nonlinearity. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Differential equations. Mathematical physics, Tome 139 (2017), pp. 44-58. http://geodesic.mathdoc.fr/item/INTO_2017_139_a4/