Eternal solutions for a reaction-diffusion equation with weighted reaction

Abstract

<p style='text-indent:20px;'>We prove existence and uniqueness of <i>eternal solutions</i> in self-similar form growing up in time with exponential rate for the weighted reaction-diffusion equation</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \partial_tu = \Delta u^m+|x|^{\sigma}u^p, $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>posed in <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{R}^N $\end{document}</tex-math></inline-formula>, with <inline-formula><tex-math id="M2">\begin{document}$ m>1 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M3">\begin{document}$ 0<p<1 $\end{document}</tex-math></inline-formula> and the critical value for the weight</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ \sigma = \frac{2(1-p)}{m-1}. $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>Existence and uniqueness of some specific solution holds true when <inline-formula><tex-math id="M4">\begin{document}$ m+p\geq2 $\end{document}</tex-math></inline-formula>. On the contrary, no eternal solution exists if <inline-formula><tex-math id="M5">\begin{document}$ m+p<2 $\end{document}</tex-math></inline-formula>. We also classify exponential self-similar solutions with a different interface behavior when <inline-formula><tex-math id="M6">\begin{document}$ m+p>2 $\end{document}</tex-math></inline-formula>. Some transformations to reaction-convection-diffusion equations and traveling wave solutions are also introduced.</p>

Self-similar blow-up patterns for a reaction-diffusion equation with weighted reaction in general dimension

“…the most interesting feature of this equation being the presence of an unbounded weight on the reaction term, which, as we shall see, has a very strong influence on the dynamics of it. In particular, the lower bound we impose σ > 2(1−p)/(m−1) is the optimal one leading to the phenomenon of finite time blow-up, which does not occur either for the non-weighted (homogeneous) equation with σ = 0 studied in [26,27,28] or even for the limiting case σ = 2(1−p)/(m−1), a case studied recently in [21] where it is shown that eternal solutions in self-similar exponential form exist.…”

“…(1.1) with σ = 0. We mention here also our recent work [21] in which the limits of the blow-up behavior are explored: it is proved there that for σ = 2(1 − p)/(m − 1), eternal solutions with exponentially fast grow-up in time exist and finite time blowup is not expected to occur for this limiting exponent. We thus deduce that the range of σ we consider is optimal with respect to finite time blow-up, and of course this illustrates the strong influence of the magnitude of σ on the dynamics of Eq.…”

Radial equivalence and applications to the qualitative theory for a class of non-homogeneous reaction-diffusion equations

“…(1.1) is mapped onto the condition L = 0 for Eq. (1.5), where L is the constant defined in (1.9), and we already know from [31,20] that L = 0 is the necessary and sufficient condition in Eq. (1.5) to ensure the existence of eternal self-similar solutions in exponential form.…”

“…(1.9) Indeed, when L > 0, backward self-similar solutions as in (1.8b) have been constructed, while if L < 0, self-similar solutions are in forward form (1.8a). Eternal, exponential selfsimilar solutions have been constructed in the special case L = 0 in [31,32,20]. We end up this presentation by introducing here a constant related to L which will be very useful in the study of self-similar solutions to Eq.…”

Radial equivalence and applications to the qualitative theory for a class of nonhomogeneous reaction‐diffusion equations

“…(1.8) Indeed, when L > 0, backward self-similar solutions as in (1.7b) have been constructed, while if L < 0, self-similar solutions are in forward form (1.7a). Eternal, exponential self-similar solutions have been constructed in the special case L = 0 in [59][60][61]. We end up this presentation by introducing here a constant related to L, which will be very useful in the study of self-similar solutions to Equation (1.1):…”