Blow-up phenomena for positive solutions of semilinear diffusion equations in a half-space: the influence of the dispersion kernel

Alfaro, Matthieu; Kavian, Otared

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 31 (2022) no. 5, pp. 1259-1286.

On considère l’équation semi linéaire ${\partial }_{t}u=Au+{\left|u\right|}^{\alpha }u$ dans le demi-espace ${ℝ}_{+}^N:={ℝ}^{N-1}\times (0,+\infty )$, où $A$ est un opérateur linéaire de diffusion, qui peut être le Laplacien « classique », ou le Laplacien fractionnaire, ou un opérateur non local et non régularisant. L’équation est complétée par une donnée initiale $u(0,x)=u_0\left(x\right)$ positive dans le demi-espace ${ℝ}_{+}^N$, et la condition de Dirichlet $u(t,x^{\prime },0)=0$ pour $x^{\prime }\in {ℝ}^{N-1}$.

On prouve que si le symbole de l’opérateur $A$ est de l’ordre de ${a\left|\xi \right|}^{\beta }$ au voisinage de l’origine $\xi =0$, avec $\beta \in (0,2]$ et $a>0$, alors toute solution strictement positive explose en temps fini dès que $0<\alpha \le \beta /(N+1)$. En revanche, on prouve l’existence de solutions strictement positives globales quand $\alpha >\beta /(N+1)$. Notons que, dans le cas considéré du demi-espace ${ℝ}_{+}^N$, l’exposant $\beta /(N+1)$ est inférieur à l’exposant de Fujita $\beta /N$ dans l’espace ${ℝ}^N$.

Ceci nous permet également de résoudre la question de l’explosion pour les solutions de l’équation semi linéaire dans tout l’espace, ${ℝ}^N$, qui sont impaires dans la direction $x_N$ (et qui, donc, changent de signe).

We consider the semilinear diffusion equation ${\partial }_{t}u=Au+{\left|u\right|}^{\alpha }u$ in the half-space ${ℝ}_{+}^N:={ℝ}^{N-1}\times (0,+\infty )$, where $A$ is a linear diffusion operator, which may be the classical Laplace operator, or a fractional Laplace operator, or an appropriate non regularizing nonlocal operator. The equation is supplemented with an initial data $u(0,x)=u_0\left(x\right)$ which is nonnegative in the half-space ${ℝ}_{+}^N$, and the Dirichlet boundary condition $u(t,x^{\prime },0)=0$ for $x^{\prime }\in {ℝ}^{N-1}$.

We prove that if the symbol of the operator $A$ is of order ${a\left|\xi \right|}^{\beta }$ near the origin $\xi =0$, for some $\beta \in (0,2]$ and $a>0$, then any positive solution of the semilinear diffusion equation blows up in finite time whenever $0<\alpha \le \beta /(N+1)$. On the other hand, we prove existence of positive global solutions of the semilinear diffusion equation in a half-space when $\alpha >\beta /(N+1)$. Notice that in the case of the half-space, the exponent $\beta /(N+1)$ is smaller than the so-called Fujita exponent $\beta /N$ in ${ℝ}^N$.

As a consequence we can also solve the blow-up issue for solutions of the above mentioned semilinear diffusion equation in the whole of ${ℝ}^N$, which are odd in the $x_N$ direction (and thus sign changing).

Classification : 35B40, 35B33, 45K05, 47G20
Mots clés : Sign changing solutions, half-space, blow-up solutions, global solutions, Fujita exponent, nonlocal diffusion, dispersal tails

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