2019
Том 71
№ 1

# 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

Abstract

We study the Cauchy problem for a degenerate parabolic equation with source and inhomogeneous density of the form $$u_t = \text{div}(\rho(x)u^{m-1}|Du|^{\lambda-1}Du) + u ^p$$ in the case where initial function decreases slowly to zero as $|x| \rightarrow \infty$. We establish conditions for the existence and nonexistence of a global-in-time solution, which substantially depend on the behavior of the initial data as $|x| \rightarrow \infty$. In the case of global solvability, we obtain an exact estimate of a solution for large times.

English version (Springer): Ukrainian Mathematical Journal 64 (2012), no. 11, pp 1698-1715.

Citation Example: Martynenko A. V., Shramenko V. N., Tedeev A. F. 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 // Ukr. Mat. Zh. - 2012. - 64, № 11. - pp. 1500-1515.

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