Qualitative analysis for a system of anisotropic parabolic equations with sign‐changing logarithmic nonlinearity

Abstract

The main purpose of this paper is to discuss the global well-posedness, asymptotic behavior, and blow-up of solutions in finite time for a parabolic system of equations driven by the (p(x), q(x))-Laplacian operator with variable exponents and sign-changing nonlinearity. More precisely, we considerwhere Ω ⊂ R N is a smooth bounded domain, N ≥ 1, p, q, s ∶ Ω → R + are continuous functions and satisfy some conditions. The main characteristic of this paper is to study the global existence, asymptotic behavior, and blow-up of solutions to the above system where the reaction terms are sign-changing and satisfy neither the monotonicity condition nor the Ambrosetti-Rabinowitz condition.

Global well‐posedness and finite time blow‐up for a class of wave equation involving fractionalp‐Laplacian with logarithmic nonlinearity

“…Furthermore, the author showed a polynomial decay estimate in the L 2 ‐norm for the solutions of this problem. Finally, we refer the reader to [36, 37] where the author has discussed the global existence, asymptotic behavior, and blow‐up of solutions for classes of parabolic equations involving the $$p(x)$$‐Laplacian with logarithmic nonlinearity. Notice that, if $$s\rightarrow 1^{-}$$ and $$q=2$$, then the equation in (1.1) reduces the following one: $$$$\begin{equation*} \hspace*{117pt}u_{tt}-\Delta _{p}u=u\log (|u|),\;\;\; \text{where}\;\;\Delta _{p}u=\text{div}(|\nabla u|^{p-2}\nabla u).\qquad\hspace*{117pt} \mathrm{{(P)}} \end{equation*}$$$$In this paper, we shall show that solving a wave equation involving the nonlinear fractional p ‐Laplacian via the Galerkin method is much easier than solving the usual quasilinear wave equations like those in ( P ).…”

“…Furthermore, the author showed a polynomial decay estimate in the 𝐿 2 -norm for the solutions of this problem. Finally, we refer the reader to [36,37] where the author has discussed the global existence, asymptotic behavior, and blow-up of solutions for classes of parabolic equations involving the 𝑝(𝑥)-Laplacian with logarithmic nonlinearity. Notice that, if 𝑠 → 1 − and 𝑞 = 2, then the equation in (1.1) reduces the following one: 𝑢 𝑡𝑡 − Δ 𝑝 𝑢 = 𝑢 log(|𝑢|), where Δ 𝑝 𝑢 = div(|∇𝑢| 𝑝−2 ∇𝑢).…”