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Dependence of the blow-up time with respect to parameters and numerical approximations for a parabolic problem. (English) Zbl 1047.35064

The problem of the behaviour of the blow-up time is investigated in this article. The problem can be described for the following semilinear parabolic problem: \[ \begin{alignedat}{2} u_t &= \lambda \Delta u + u^p , &&\quad\text{in } \Omega \times (0,T),\\ u &= 0, &&\quad\text{on } \partial \Omega \times (0,T),\\ u(x,0) &= u_0(x)= \phi(x) + h f(x) , &&\quad x \in \Omega. \end{alignedat} \] for \( p>1, \;\lambda >0\) and \( h \in \mathbb{R} \) are parameters, \( \Omega\) is a bounded smooth domain in \(\mathbb{R}\)\(^n \) and \(u_0\) is regular in order to guarantee existence, uniqueness and regularity of the solution. It is well known, that the reaction term in the equation of power type for \(p>1\) causes that blow-up phenomenon that is there exists a finite time \(T\), so called blow-up time, in which the \(L_\infty\) norm of the solution tends to infinity. The authors are interested in the study of the dependence of the blow-up time \(T\) on the parameters that appears in the problem: \( p, \lambda, h\). They improve the previous results, that blow-up time is continuous with respect to the initial data, showing the existence of a modulus of continuity for \(T\) not only with respect to the initial condition but also with respect to the diffusion coefficient \(\lambda\) and the reaction power \(p\).
In the second part of the article the authors study the behaviour of the blow-up time for numerical approximations for \( \lambda=1\) and some initial condition. They improve the results that numerical blow-up time converges to the continuous one as the mesh parameter \(h\) tends to zero by the rate of convergence of the form \( | T -T_h| \leq C h^\gamma,\) where \(T_h\) is numerical blow-up time, \(C\) and \(\gamma\) are positive constants.

MSC:

35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35K55 Nonlinear parabolic equations
35B65 Smoothness and regularity of solutions to PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35B40 Asymptotic behavior of solutions to PDEs
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