Self-similar solutions preventing finite time blow-up for reaction-diffusion equations with singular potential

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

“…This transformation generalizes the one introduced in [60, Section 6], and we notice that it is expected that the condition $$$ L\left({\sigma}_1,{\sigma}_2\right)=0 $$$ be the critical connection between exponents allowing for eternal solutions of the form (). In fact, if we go back to the transformation () and we recall the value of $$$ \sigma $$$ from (), we notice that $$$$ \sigma \left(m-1\right)+2\left(p-1\right)=\frac{2L\left({\sigma}_1,{\sigma}_2\right)}{\sigma_1+{\sigma}_2}, $$$$ which suggests that the condition $$$ L\left({\sigma}_1,{\sigma}_2\right)=0 $$$ for Equation () is mapped onto the condition $$$ L=0 $$$ for Equation (), where $$$ L $$$ is the constant defined in (), and we already know from [59, 60] that $$$ L=0 $$$ is the necessary and sufficient condition in Equation () to ensure the existence of eternal self‐similar solutions in exponential form. Notice also that the critical case $$$ {\sigma}_1=-2 $$$ (its criticality for the nonhomogeneous porous medium equation follows, e.g., from works such as [13–15]) works very well with this transformation by simply letting $$$ \delta =0 $$$…”

“…The analysis of Equation () in this range of exponents was practically lacking from theory and some significant recent progress in its understanding has been achieved by the authors in a series of papers; see, for example, [51, 52, 54–56, 59], where different ranges related to the dimension $$$ N $$$ and the sign of the constant $$$ L $$$ in () were considered. The outcome of this analysis was quite unexpected, all the exponents having a strong influence in both the form of the self‐similar solutions and of their profiles.…”

“…Proof Our starting point is the statement of [52, Theorem 1.1], ensuring that there exists a unique radially symmetric self‐similar solution in the form () to Equation () having a compactly supported profile, with self‐similar exponents $$$$ \overline{\alpha}=-\frac{\sigma +2}{\sigma \left(m-1\right)+2\left(p-1\right)},\kern2em \overline{\beta}=-\frac{m-p}{\sigma \left(m-1\right)+2\left(p-1\right)} $$$$ and local behavior as $$$ \xi \to 0 $$$ given by $$$$ \overline{f}\left(\overline{\xi}\right)\sim {\left[D\left(\sigma \right)-\frac{m-p}{m\left(\overline{N}+\sigma \right)\left(\sigma +2\right)}{\overline{\xi}}^{\sigma +2}\right]}^{1/\left(m-p\right)}, $$$$ where we use the notation with bar for the variables related to Equation (), provided that $$$ 1\le p<1-\sigma \left(m-1\right)/2 $$$. We apply our transformation ()–(), and we infer from the value of $$$ \sigma $$$ in () that the self‐similar exponents are mapped to …”

“…More recently, the authors developed a larger project of understanding the dynamics of Equation () starting from the classification of its self‐similar solutions. Focusing at first on the range of exponents $$$ 1<p<m $$$, we have shown that both the occurrence of the finite time blowup, its sets and rates, and the geometric form and evolution of the self‐similar profiles depend strongly on the magnitude of $$$ \sigma \ge -2 $$$; see, for example, the results in [51–56] and references therein. The very interesting case $$$ p=m $$$ is critical, and Equation () has been considered with $$$ p=m $$$ in [57, 58].…”

“…Indeed, when $$$ L>0 $$$, backward self‐similar solutions as in () have been constructed, while if $$$ L<0 $$$, self‐similar solutions are in forward form (). Eternal, exponential self‐similar solutions have been constructed in the special case $$$ L=0 $$$ in [59–61]. 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 (): $$$$ L\left({\sigma}_1,{\sigma}_2\right):= {\sigma}_2\left(m-1\right)+2\left(p-1\right)-{\sigma}_1\left(m-p\right).$$…”