Blow-up results for a strongly perturbed semilinear heat equation: theoretical analysis and numerical method.

*(English)*Zbl 1334.35148The blow-up behavior of classical solutions of the following initial value problem for the semilinear heat equation in \(\mathbb{R}^n\):
\[
\begin{aligned} &\partial_tu=\Delta u+|u|^{p-1}u+h(u), \\ &u|_{t=0}=u_0\in L^\infty(\mathbb{R}^n), \end{aligned}\tag{1}
\]
is investigated, where \(p\) is subcritical (i.e., \(p<\frac{n+2}{n-2}\) if \(n\geq 3\)) and
\[
h(u)=\mu\frac{|u|^{p-1}u}{\log_a(2+u^2)},\qquad a>0,\;\mu\in\mathbb{R}.
\]
Similarity variables are introduced and the existence of a Lyapunov functional for the resulting equation is shown, which allows to determine the blow-up rate and blow-up limit of solutions. Different from the classical case, where \(h\equiv 0\), problem (1) is not scaling invariant and the PDE written in similarity variables is non-autonomous, which requires to adapt previous techniques.

The asymptotic behavior of solutions approaching a blow-up point is further investigated. The blow-up behavior is classified and explicit blow-up profiles are determined. The article extends the findings obtained for solutions of (1) with \(a>1\) in [V. T. Nguyen, Discrete Contin. Dyn. Syst. 35, No. 8, 3585–3626 (2015; Zbl 1327.35195)] to values \(a\in \;]0,1]\). The corresponding results for the classical problem with \(h\equiv 0\) have been obtained in [J. J. L. Velázquez, Commun. Partial Differ. Equations 17, No. 9–10, 1567–1596 (1992; Zbl 0813.35009)].

Finally, numerical simulations are presented to illustrate the results. The blow-up profile for solutions is computed (in a bounded one-dimensional domain) by a mesh-refining algorithm based on the rescaling method in [M. Berger and R. V. Kohn, Commun. Pure Appl. Math. 41, No. 6, 841–863 (1988; Zbl 0652.65070)].

The asymptotic behavior of solutions approaching a blow-up point is further investigated. The blow-up behavior is classified and explicit blow-up profiles are determined. The article extends the findings obtained for solutions of (1) with \(a>1\) in [V. T. Nguyen, Discrete Contin. Dyn. Syst. 35, No. 8, 3585–3626 (2015; Zbl 1327.35195)] to values \(a\in \;]0,1]\). The corresponding results for the classical problem with \(h\equiv 0\) have been obtained in [J. J. L. Velázquez, Commun. Partial Differ. Equations 17, No. 9–10, 1567–1596 (1992; Zbl 0813.35009)].

Finally, numerical simulations are presented to illustrate the results. The blow-up profile for solutions is computed (in a bounded one-dimensional domain) by a mesh-refining algorithm based on the rescaling method in [M. Berger and R. V. Kohn, Commun. Pure Appl. Math. 41, No. 6, 841–863 (1988; Zbl 0652.65070)].

Reviewer: Stefanie Sonner (Kaiserslautern)