ASYMPTOTIC BEHAVIOR OF THE BLOW-UP AND GLOBAL EXISTENCE FOR A PARABOLIC EQUATION INVOLVING A CRITICAL EXPONENT
Keywords:
semi-linear diffusion equation, blow-up, continuity, numerical blow-up time, discretizations, semidiscretizations, convergence, asymptotic behavior, global existence, finite difference methodDOI:
https://doi.org/10.17654/0975045223002Abstract
Considering the energy critical of semi-linear diffusion equation for the following initial-boundary value problem:
$$
(P) \begin{cases}u_t(x, t)-L u(x, t)-e^{b t}(u(x, t))^{p(x)}=0 & \text { in } \Omega \times(0, T), \\ u(x, t)=0 & \text { on } \partial \Omega \times(0, T), \\ u(x, 0)=u_0(x) & \text { in } \Omega,\end{cases}
$$
we deal with the blow-up of the solutions to a semi-linear heat equation with a reaction given by variable sources, where the nonlinear terms are of power type functions, with zero Dirichlet boundary condition and nonnegative initial datum.
We use explicit linear and implicit Euler finite difference schemes with a special time-step formula to compute the blow-up solutions, and to estimate the blow-up times. The asymptotic behavior of solutions near their singularities is only completely understood in the special case where the source is a power and using matching asymptotic procedures for a large value of p, and the corresponding regime displays a polynomial type blow-up speed.
We prove that every numerical solution blows up in finite time if p > 1 and that the numerical blow-up time converges to the continuous one as the mesh parameter goes to zero.
Some conditions to parameters and exponents of sources are given to obtain lower-upper bounds for the time of blow-up and some global existence results of the problem (P). We also show that the solution blows up in a finite time and its blow-up time as a function of the initial datum is continuous.
Finally, we carry out the numerical simulations to the discrete graphs obtained from using these methods to support the numerical results and to confirm some known blow-up properties for the studied problem.
Received: March 15, 2022
Accepted: May 5, 2022
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