Blow Up Profiles for a Quasilinear Reaction–Diffusion Equation with Weighted Reaction with Linear Growth


We study the blow up profiles associated to the following second order reaction–diffusion equation with non-homogeneous reaction:

$$\begin{aligned} \partial _tu=\partial _{xx}(u^m) + |x|^{\sigma }u, \end{aligned}$$

with \(\sigma >0\). Through this study, we show that the non-homogeneous coefficient \(|x|^{\sigma }\) has a strong influence on the blow up behavior of the solutions. First of all, it follows that finite time blow up occurs for self-similar solutions u, a feature that does not appear in the well known autonomous case \(\sigma =0\). Moreover, we show that there are three different types of blow up self-similar profiles, depending on whether the exponent \(\sigma \) is closer to zero or not. We also find an explicit blow up profile. The results show in particular that global blow up occurs when \(\sigma >0\) is sufficiently small, while for \(\sigma >0\) sufficiently large blow up occurs only at infinity, and we give prototypes of these phenomena in form of self-similar solutions with precise behavior. This work is a part of a larger program of understanding the influence of non-homogeneous weights on the blow up sets and rates.

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R.I. is supported by the ERC Starting Grant GEOFLUIDS 633152. A.S. is partially supported by the Spanish Project MTM2017-87596-P.

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Correspondence to Razvan Gabriel Iagar.

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Iagar, R.G., Sánchez, A. Blow Up Profiles for a Quasilinear Reaction–Diffusion Equation with Weighted Reaction with Linear Growth. J Dyn Diff Equat 31, 2061–2094 (2019).

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  • Reaction–diffusion equations
  • Non-homogeneous reaction
  • Blow up
  • Critical case
  • Self-similar solutions
  • Phase space analysis

Mathematics Subject Classification

  • 35B33
  • 35B40
  • 35K10
  • 35K67
  • 35Q79