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We study the propagation properties of the reaction-diffusion equation of Fisher type u t l (u m−" u x) x ju p (1ku) for x ? , t 0, with p 1 m. Taking into account that solutions of the Cauchy problem are nonunique if mjp 2, we prove that the minimal solutions in this case tend to propagate with certain minimal speed cJ(m, p). More precisely, if we translate any solution with a velocity QcQ cJ, we get the limit in time value one, and if the initial value u(:, 0) vanishes, say, for x 0, the minimal solution translated with velocity c cJ tends to zero. Also, an interface appears for the minimal solutions, whose asymptotic velocity is cJ. This behaviour depends upon the existence of special solutions of travelling wave form. Travelling waves have been widely studied for diffusion equations related with the above. We characterize here the minimal velocity cJ for which travelling waves exist, as an analytic function of the parameters m and p, for every mjp 2, by viewing it as an anomalous exponent. Some local properties of the minimal solutions and their interfaces in the case mjp 2 are also proved.

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We study the self-similar blow-up profiles associated to the following second-order reaction-diffusion equation with strong weighted reaction and unbounded weight:\partial_{t}u=\partial_{xx}(u^{m})+|x|^{\sigma}u^{p},posed for {x\in\mathbb{R}}, {t\geq 0}, where {m>1}, {0<p<1} and {\sigma>\frac{2(1-p)}{m-1}}. As a first outcome, we show that finite time blow-up solutions in self-similar form exist for {m+p>2} and σ in the considered range, a fact that is completely new: in the already studied reaction-diffusion equation without weights there is no finite time blow-up when {p<1}. We moreover prove that, if the condition {m+p>2} is fulfilled, all the self-similar blow-up profiles are compactly supported and there exist two different interface behaviors for solutions of the equation, corresponding to two different interface equations. We classify the self-similar blow-up profiles having both types of interfaces and show that in some cases global blow-up occurs, and in some other cases finite time blow-up occurs only at space infinity. We also show that there is no self-similar solution if {m+p<2}, while the critical range {m+p=2} with {\sigma>2} is postponed to a different work due to significant technical differences.

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<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&gt;1 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M3">\begin{document}$ 0&lt;p&lt;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&lt;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&gt;2 $\end{document}</tex-math></inline-formula>. Some transformations to reaction-convection-diffusion equations and traveling wave solutions are also introduced.</p>

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