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Abstract

The aim of this paper is to consider the following fractional parabolic problem $\begin{array}{l} \{\begin{matrix} u_{t} + {(- Δ)}_{p}^{α} u + {(- Δ)}_{q}^{β} u = f (x, u) & (x, t) \in Ω \times (0, \infty), \\ u = 0 & (x, t) \in (R^{N} ∖ Ω) \times (0, \infty), \\ u (x, 0) = u_{0} (x) & x \in Ω, \end{matrix} \end{array}$ where $Ω \subset R^{N}$ is a bounded domain with Lipschitz boundary, ${(- Δ)}_{p}^{α}$ is the fractional p-Laplacian with $0 < α < 1 < p < \infty$ , ${(- Δ)}_{q}^{β}$ is the fractional q-Laplacian with $0 < β < α < 1 < q < p < \infty$ , $r > 1$ and $λ > 0$ . The global existence of nonnegative solutions is obtained by combining the Galerkin approximations with the potential well theory. Then, by virtue of a differential inequality technique, we give a decay estimate of solutions.

Keywords

global existence decay estimates

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Global existence and asymptotic behavior of solutions to fractional ( p,q ) -Laplacian equations

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