On the behavior of solutions of the Cauchy problem for a degenerate parabolic equation with source in the case where the initial function slowly vanishes. (English. Ukrainian original) Zbl 1273.35167
Ukr. Math. J. 64, No. 11, 1698-1715 (2013); translation from Ukr. Mat. Zh. 64, No. 11, 1500-1515 (2012).
Summary: We study the Cauchy problem for a degenerate parabolic equation with source and inhomogeneous density of the form
\[
u_t=\mathrm{div}(\rho(x)u^{m-1}| Du|^{\lambda-1}Du)+u^p
\]
in the case where the initial function slowly vanishes as \(| x|\rightarrow\infty\). We establish conditions for the existence and nonexistence of a global (in time) solution. These conditions strongly depend on the behavior of the initial data as \(| x|\rightarrow\infty\). In the case of global solvability, we establish a sharp estimate of the solution for large times.
MSC:
| 35K65 | Degenerate parabolic equations |
| 35K15 | Initial value problems for second-order parabolic equations |
Keywords:
global existence and nonexistenceReferences:
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