On blow-up and asymptotic behavior of solutions to a nonlinear parabolic equation of second order with nonlinear boundary conditions

summary:We obtain some sufficient conditions under which solutions to a nonlinear parabolic equation of second order with nonlinear boundary conditions tend to zero or blow up in a finite time. We also give the asymptotic behavior of solutions which tend to zero as $t\rightarrow\infty$. Finally, we obtain the asymptotic behavior near the blow-up time of certain blow-up solutions and describe their blow-up set

Asymptotic behavior for a nonlocal diffusion problem with Neumann boundary conditions and a reaction term

In this paper, we consider the following initial value problem u_t(x,t)=\int_{\Omega}J(x-y)(u(y,t)-u(x,t))dy-\gamma u^{p}(x,t)& \mbox{in}& \overline{\Omega}\times(0,\infty),u(x,0)=u_{0}(x)>0& \mbox{in}& \overline{\Omega},            where $\gamma\ in \{-1,1\}$ is a parameter, $\Omega$ is a bounded domain in      $\mathbb{R}^{N}$ with smooth boundary $\partial\Omega$, p>1, $J$:      $\mathbb{R}^N\longrightarrow\mathbb{R}$ is a kernel which is nonnegative,      measurable, symmetric, bounded and $\int_{\mathbb{R}^N}J(z)dz=1,$ the      initial datum $u_0 \ in C^0(\overline{\Omega})$, u_0(x)>0 in      $\overline{\Omega}$. We show that, if $\gamma=1$, then the solution $u$      of the above problem tends to zero as $t\rightarrow\infty$ uniformly in      $x\in\overline{\Omega}$, and a description of its asymptotic behavior is      given. We also prove that, if $\gamma=-1$, then the solution $u$ blows up      in a finite time, and its blow-up time goes to that of the solution of a      certain ODE as the $L^{\infty}$ norm of the initial datum goes to      infinity

Continuity of the quenching time in a semilinear heat equation with Neumann boundary condition

This paper concerns the study of a semilinear parabolic equation subject to Neumann boundary conditions and positive initial datum. Under some assumptions, we show that the solution of the above problem quenches in a finite time and estimate its quenching time. We also prove the continuity of the quenching time as a function of the initial datum. Finally, we give some numerical results to illustrate our analysis

Extinction time for some nonlinear heat equations

This paper concerns the study of the extinction time of the solution of the following initial-boundary value problem [left{% begin{array}{ll} hbox{$u_t=varepsilon Lu(x,t)-f(u)quad mbox{in}quad Omegatimesmathbb{R}_{+}$,} \ hbox{$u(x,t)=0quad mbox{on}quadpartialOmegatimesmathbb{R}_{+}$,} \ hbox{$u(x,0)=u_{0}(x)>0quad mbox{in}quad Omega$,} \ end{array}%right. ] where $Omega$ is a bounded domain in $mathbb{R}^{N}$ with smooth boundary $partialOmega$, $varepsilon$ is a positive parameter, $f(s)$ is a positive, increasing, concave function for positive values of s, $f(0)=0$, $int_{0}frac{ds}{f(s)}<+infty$, $L$ is an elliptic operator. We show that the solution of the above problem extincts in a finite time and its extinction time goes to that of the solution $alpha(t)$ of the following differential equation [alpha^{\u27}(t)=-f(alpha(t)),quad t>0,quad alpha(0)=M,] as $varepsilon$ goes to zero, where $M=sup_{xin Omega}u_{0}(x)$. We also extend the above result to other classes of nonlinear parabolic equations. Finally, we give some numerical results to illustrate our analysis

Quenching time of some nonlinear wave equations

summary:In this paper, we consider the following initial-boundary value problem {\left\rbrace \begin{array}{ll} u_{tt}(x,t)=\varepsilon Lu(x,t)+f\big (u(x,t)\big )\quad \mbox {in}\quad \Omega \times (0,T)\,,\\ u(x,t)=0 \quad \mbox {on}\quad \partial \Omega \times (0,T)\,, \\ u(x,0)=0 \quad \mbox {in}\quad \Omega \,, \\ u_t(x,0)=0 \quad \mbox {in}\quad \Omega \,, \end{array}\right.} where $\Omega$ is a bounded domain in $\mathbb{R}^N$ with smooth boundary $\partial \Omega$, $L$ is an elliptic operator, $\varepsilon$ is a positive parameter, $f(s)$ is a positive, increasing, convex function for $s\in (-\infty ,b)$, $\lim _{s\rightarrow b}f(s)=\infty$ and $\int _0^b\frac{ds}{f(s)}0$. Under some assumptions, we show that the solution of the above problem quenches in a finite time and its quenching time goes to that of the solution of the following differential equation $${\left\rbrace \begin{array}{ll} \alpha ^{\prime \prime }(t)=f(\alpha (t))\,,&\quad t>0\,, \\ \alpha (0)=0\,,\quad \alpha ^{\prime }(0)=0\,, \end{array}\right.}$$ as $\varepsilon$ goes to zero. We also show that the above result remains valid if the domain $\Omega$ is large enough and its size is taken as parameter. Finally, we give some numerical results to illustrate our analysis