Nonexistence of solutions of certain semilinear heat equations

Durvudkhan Suragan; Bharat Talwar

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arXiv · 2210.11307

Nonexistence of solutions of certain semilinear heat equations

Durvudkhan Suragan·Bharat Talwar

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Abstract

We consider a semilinear heat equation involving a forcing term which depends only on the space variable. To start with, the existence of a local mild solution is proved through an application of the Banach fixed-point theorem. With the help of carefully defined test functions, we then prove the nonexistence of global weak solutions. The most crucial step is to find the function $d(x)$ used in our proofs, which seems to depends only upon the considered vector fields. This leads to lower bounds for a possible critical Fujita-type exponent. The same function $d(x)$ could lead to a potential norm function which would be most suitable while working with these vector fields. Section 4 is the attraction of this paper in which we apply our approach to all of the vector fields discussed by Biagi, Bonfiglioli and Bramanti, giving rise to Grushin-type and Engel-type PDOs, and more. An upper bound for the blow-up time of local solutions is also provided in each of these cases.

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Durvudkhan Suragan, Bharat Talwar. 2022-10-20. Nonexistence of solutions of certain semilinear heat equations. https://arxiv.org/abs/2210.11307

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