## Blow-up results for a strongly perturbed semilinear heat equation: theoretical analysis and numerical method.(English)Zbl 1334.35148

The 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)].

### MSC:

 35K58 Semilinear parabolic equations 35B44 Blow-up in context of PDEs 35K15 Initial value problems for second-order parabolic equations 35K10 Second-order parabolic equations 35B40 Asymptotic behavior of solutions to PDEs

### Citations:

Zbl 1327.35195; Zbl 0813.35009; Zbl 0652.65070
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