Stabilization of the critical semilinear wave equation with Dirichlet-Neumann boundary condition on bounded domain

2021 ◽  
pp. 125610
Author(s):  
Hao Li ◽  
Zhen-Hu Ning ◽  
Fengyan Yang
Keyword(s):  
Boundary Condition ◽  
Wave Equation ◽  
Bounded Domain ◽  
Neumann Boundary Condition ◽  
Neumann Boundary ◽  
Semilinear Wave Equation

scholarly journals Blow-up for the wave equation with nonlinear source and boundary damping terms

2015 ◽  
Vol 145 (4) ◽  
pp. 759-778 ◽  
Author(s):  
Alessio Fiscella ◽  
Enzo Vitillaro
Keyword(s):  
Boundary Condition ◽  
Wave Equation ◽  
Initial Data ◽  
Neumann Boundary Condition ◽  
Blow Up ◽  
Neumann Boundary ◽  
Nonlinear Dissipation ◽  
Typical Problem ◽  
Nonlinear Source ◽  
Boundary Damping

The paper deals with blow-up for the solutions of an evolution problem consisting in a semilinear wave equation posed in a boundedC1,1open subset of ℝn, supplied with a Neumann boundary condition involving a nonlinear dissipation. The typical problem studied iswhere∂Ω=Γ0∪Γ1,Γ0∩Γ1= ∅,σ(Γ0) > 0, 2 <p≤ 2(n− 1)/(n− 2) (whenn≥ 3),m> 1,α∈L∞(Γ1),α≥ 0 andβ≥ 0. The initial data are posed in the energy space.The aim of the paper is to improve previous blow-up results concerning the problem.

scholarly journals Multiplicity Results for thepx-Laplacian Equation with Singular Nonlinearities and Nonlinear Neumann Boundary Condition

2016 ◽  
Vol 2016 ◽  
pp. 1-14 ◽  
Author(s):  
K. Saoudi ◽  
M. Kratou ◽  
S. Alsadhan
Keyword(s):  
Boundary Condition ◽  
Neumann Problem ◽  
Bounded Domain ◽  
Neumann Boundary Condition ◽  
Neumann Boundary ◽  
Positive Parameter ◽  
Multiplicity Results ◽  
Laplacian Equation ◽  
Variational Techniques ◽  
Singular Nonlinearities

We investigate the singular Neumann problem involving thep(x)-Laplace operator:Pλ{-Δpxu+|u|px-2u  =1/uδx+fx,u, in  Ω;  u>0,  in  Ω;  ∇upx-2∂u/∂ν=λuqx,  on  ∂Ω}, whereΩ⊂RNN≥2is a bounded domain withC2boundary,λis a positive parameter, andpx,qx,δx, andfx,uare assumed to satisfy assumptions(H0)–(H5)in the Introduction. Using some variational techniques, we show the existence of a numberΛ∈0,∞such that problemPλhas two solutions forλ∈0,Λ,one solution forλ=Λ, and no solutions forλ>Λ.

scholarly journals $W^{1,N}$ versus $C^1$ local minimizer for a singular functional with Neumann boundary condition

2017 ◽  
Vol 37 (1) ◽  
pp. 71
Author(s):  
Kamel Saoudi
Keyword(s):  
Boundary Condition ◽  
Bounded Domain ◽  
Neumann Boundary Condition ◽  
Multiple Solutions ◽  
Local Minimum ◽  
Smooth Boundary ◽  
Lagrange Equation ◽  
Neumann Boundary ◽  
Local Minimizer ◽  
Multiplicity Of Positive Solutions

Let $\Omega\subset\R^N,$ be a bounded domain with smooth boundary. Let $g:\R^+\to\R^+$ be a continuous on $(0,+\infty)$ non-increasing and satisfying $$c_1=\liminf_{t\to 0^+}g(t)t^{\delta}\leq\underset{t\to 0^+}{\limsup} g(t)t^{\delta}=c_2,$$ for some $c_1,c_2>0$ and $0<\delta<1.$ Let $f(x,s) = h(x,s)e^{bs^{\frac{N}{N-1}}},$ $b>0$ is a constant.Consider the singular functional $I: W^{1,N}(\Omega)\to \R$ defined as \begin{eqnarray*}&&I(u)\eqdef\frac{1}{N}\|u\|^N_{W^{1,N}(\Omega)}-\int_{\Omega}G(u^+)\,{\rm d} x-\int_{\Omega}F(x,u^+) \,{\rm d} x\nonumber\\&& -\frac{1}{q+1}||u||^{q+1}_{L^{q+1}(\partial\Omega)}\nonumber\end{eqnarray*} where $F(x,u)=\int_0^sf(x,s)\,{\rm d}s$, $G(u)=\int_0^s g(s)\,{\rm d}s$. We show that if $u_0\in C^1(\overline{\Omega})$ satisfying $u_0\geq \eta \mbox{dist}(x,\partial\Omega)$, for some $0<\eta$, is a local minimum of $I$ in the $C^1(\overline{\Omega})\cap C_0(\overline{\Omega})$ topology, then it is also a local minimum in $W^{1,N}(\Omega)$ topology. This result is useful %for proving multiple solutions to the associated Euler-lagrange equation ${\rm (P)}$ defined below.to prove the multiplicity of positive solutions to critical growth problems with co-normalboundary conditions.

scholarly journals Deriving The Upper Blow-up Rate Estimate for a Parabolic Problem

2020 ◽  
pp. 200-203
Author(s):  
Maan A. Rasheed
Keyword(s):  
Boundary Condition ◽  
Heat Equation ◽  
Bounded Domain ◽  
Neumann Boundary Condition ◽  
Parabolic Problem ◽  
Blow Up ◽  
Neumann Boundary ◽  
Rate Estimate ◽  
Blow Up Rate

In this paper, the blow-up solutions for a parabolic problem, defined in a bounded domain, are studied. Namely, we consider the upper blow-up rate estimate for heat equation with a nonlinear Neumann boundary condition defined on a ball in Rn.

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