Blow-up for Semidiscretizations of some Semilinear Parabolic Equations with a Convection Term

This paper concerns the study of the numerical approximation for the following parabolic equations with a convection termwhere p > 1.We obtain some conditions under which the solution of the semidiscrete form of the above problem blows up in a finite time and estimate its semidiscrete blow-up time. We also prove that the semidiscrete blow-up time converges to the real one, when the mesh size goes to zero. Finally, we give some numerical experiments to illustrate ours analysis

Numerical quenching solutions of localized semilinear parabolic equation

This paper concerns the study of the numerical approximationfor the following initial-boundary value problem:ut(x; t) = uxx(x; t) + E(1 - u(0; t))-p; (x; t) 2 (-l; l) x (0; T),u(-l; t) = 0; u(l; t) = 0; t in (0; T),u(x; 0) = u0(x) and gt;= 0; x in (-l; l),where p and gt; 1, l = 1/2 and E and gt; 0. Under some assumptions, we prove that the solution of a semidiscrete form of the above problem quenches in a nite time and estimate its semidiscrete quenching time. We also show that the semidiscrete quenching time in certain cases converges to the real one when the mesh size tends to zero. Finally,we give some numerical experiments to illustrate our analysis

Numerical quenching for a semilinear parabolic equation

This paper concerns the study of the numerical approximation for the nonlinear parabolic boundary value problem with the source term leading to the quenching in finite time. We find some conditions under which the solution of a semidiscrete form of the above problem quenches in a finite time and estimate its semidiscrete quenching time. We also prove that the semidiscrete quenching time converges to the real one when the mesh size goes to zero. A similar study has been also investigated taking a discrete form of the above problem. Finally, we give some numerical experiments to illustrate our analysis. First Published Online: 14 Oct 201

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

Asymptotic behavior of solutions to nonlinear parabolic equation with nonlinear boundary conditions

We show that solutions of a nonlinear parabolic equation of second order with nonlinear boundary conditions approach zero as t approaches infinity. Also, under additional assumptions, the solutions behave as a function determined here