A complete characterisation of local existence for semilinear heat equations in Lebesgue spaces

Laister, R.; Robinson, J.C.; Sierżęga, M.; Vidal-López, A.

doi:10.1016/j.anihpc.2015.06.005

Laister, R. ; Robinson, J.C. ; Sierżęga, M. ; Vidal-López, A.

Annales de l'I.H.P. Analyse non linéaire, Tome 33 (2016) no. 6, pp. 1519-1538

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Résumé

We consider the scalar semilinear heat equation $u_{t} - Δ u = f (u)$ , where $f : [0, \infty) \to [0, \infty)$ is continuous and non-decreasing but need not be convex. We completely characterise those functions f for which the equation has a local solution bounded in $L^{q} (Ω)$ for all non-negative initial data $u_{0} \in L^{q} (Ω)$ , when $Ω \subset R^{d}$ is a bounded domain with Dirichlet boundary conditions. For $q \in (1, \infty)$ this holds if and only if ${\lim \sup}_{s \to \infty} s^{- (1 + 2 q / d)} f (s) < \infty$ ; and for $q = 1$ if and only if $\int_{1}^{\infty} s^{- (1 + 2 / d)} F (s) d s < \infty$ , where $F (s) = \sup_{1 \leq t \leq s} f (t) / t$ . This shows for the first time that the model nonlinearity $f (u) = u^{1 + 2 q / d}$ is truly the ‘boundary case’ when $q \in (1, \infty)$ , but that this is not true for $q = 1$ .

The same characterisations hold for the equation posed on the whole space $R^{d}$ provided that ${\lim \sup}_{s \to 0} f (s) / s < \infty$ .

Zbl

DOI : 10.1016/j.anihpc.2015.06.005

Keywords: Semilinear heat equation, Dirichlet problem, Local existence, Non-existence, Instantaneous blow-up, Dirichlet heat kernel

@article{AIHPC_2016__33_6_1519_0,
     author = {Laister, R. and Robinson, J.C. and Sier\.z\k{e}ga, M. and Vidal-L\'opez, A.},
     title = {A complete characterisation of local existence for semilinear heat equations in {Lebesgue} spaces},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {1519--1538},
     publisher = {Elsevier},
     volume = {33},
     number = {6},
     year = {2016},
     doi = {10.1016/j.anihpc.2015.06.005},
     zbl = {1349.35169},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2015.06.005/}
}

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Laister, R.; Robinson, J.C.; Sierżęga, M.; Vidal-López, A. A complete characterisation of local existence for semilinear heat equations in Lebesgue spaces. Annales de l'I.H.P. Analyse non linéaire, Tome 33 (2016) no. 6, pp. 1519-1538. doi: 10.1016/j.anihpc.2015.06.005

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