2 research outputs found
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{,} \
hbox{,} \
hbox{,} \
end{array}%right. ]
where is a bounded domain in with smooth boundary , is a positive parameter, is a positive, increasing, concave function for positive values of s, , , 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 of the following differential equation
[alpha^{\u27}(t)=-f(alpha(t)),quad t>0,quad alpha(0)=M,] as
goes to zero, where .
We also extend the above result to other classes of nonlinear
parabolic equations. Finally, we give some numerical results to
illustrate our analysis
Numerical Blow-Up Time for a Semilinear Parabolic Equation with Nonlinear Boundary Conditions
We obtain some conditions under which the positive solution for
semidiscretizations of the semilinear equation ut=uxx−a(x,t)f(u), 0<x<1, t∈(0,T), with boundary conditions ux(0,t)=0, ux(1,t)=b(t)g(u(1,t)), blows up in a finite time and estimate its semidiscrete blow-up time. We also establish
the convergence of the semidiscrete blow-up time and obtain some results about
numerical blow-up rate and set. Finally, we get an analogous result taking
a discrete form of the above problem and give some computational results to
illustrate some points of our analysis