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Well‐posedness of solutions for a class of quasilinear wave equations with strong damping and logarithmic nonlinearity.

Well‐posedness of solutions for a class of quasilinear wave equations with strong damping and logarithmic nonlinearity.

Authors :
Ding, Hang
Zhou, Jun
Source :
Studies in Applied Mathematics. Aug2022, Vol. 149 Issue 2, p441-486. 46p.
Publication Year :
2022

Abstract

This paper investigates the well‐posedness of solutions for the following quasilinear wave equation with strong damping and logarithmic nonlinearity in a bounded domain with homogeneous Dirichlet boundary: utt−Δu−div(∇u1+|∇u|2)−Δut=|u|r−2ulog|u|$u_{tt}-\Delta u-{\rm div}(\frac{\nabla u}{\sqrt {1+|\nabla u|^2}})-\Delta u_t=|u|^{r-2}u\log |u|$, where r≥2$r\ge 2$. By virtue of the classical Faedo–Galerkin method and some technical efforts, we first establish the local well‐posedness of solutions. Then we discuss the dynamical behaviors of solutions in detail: 1.When r≥2$r\ge 2$ and I(u0)>0$I(u_0)>0$, we show that the solutions exist globally with subcritical and critical initial energy, where I(u0)$I(u_0)$ denotes the Nehari functional with the initial value u0. Especially, under further suitable assumptions about the initial data, we show that the energy functional decays exponentially. 2.When r>2$r>2$ and I(u0)<0$I(u_0)<0$, we show that the solutions blow up in finite time with subcritical and critical initial energy. Moreover, by removing the restriction I(u0)<0$I(u_0)<0$, we prove that the solutions may blow up in finite time with arbitrary high initial energy. In particular, we derive the upper and lower bounds of the blow‐up time. 3.When r=2$r=2$ and I(u0)<0$I(u_0)<0$, we show that the maximal existence time of solutions can be extended to infinity and the solutions blow up at infinity with subcritical, critical, and arbitrary high initial energy. [ABSTRACT FROM AUTHOR]

Details

Language :
English
ISSN :
00222526
Volume :
149
Issue :
2
Database :
Academic Search Index
Journal :
Studies in Applied Mathematics
Publication Type :
Academic Journal
Accession number :
158479734
Full Text :
https://doi.org/10.1111/sapm.12498