Geodesic
Weak solutions of second-order quasilinear parabolic equations with double non-linearity
Sbornik. Mathematics, Tome 188 (1997) no. 9, pp. 1343-1370

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The first boundary-value problem for the equation $$ \beta (u)\frac {\partial u}{\partial t}-\sum _{i=1}^nD_iA_i(t,x,u,Du)+ A_0(t,x,u,Du)=0 $$ is considered in a bounded subdomain of $n$. The function $\beta (u)$ is assumed to be continuous and satisfy the following growth conditions: $$ c|u|^{r-2}\leqslant \beta (u)\leqslant C\bigl (|u|^{r-2}+1\bigr ),\qquad r\geqslant 2. $$ The other coefficients satisfy the standard conditions of the theory of monotone operators. An existence theorem for a global weak solution is proved. 
@article{SM_1997_188_9_a4,
     author = {G. I. Laptev},
     title = {Weak solutions of second-order quasilinear parabolic equations with double non-linearity},
     journal = {Sbornik. Mathematics},
     pages = {1343--1370},
     publisher = {mathdoc},
     volume = {188},
     number = {9},
     year = {1997},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1997_188_9_a4/}
}
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G. I. Laptev. Weak solutions of second-order quasilinear parabolic equations with double non-linearity. Sbornik. Mathematics, Tome 188 (1997) no. 9, pp. 1343-1370. http://geodesic.mathdoc.fr/item/SM_1997_188_9_a4/