Advances in Pure and Applied Mathematics
, Volume 10 (4): 14 – Oct 1, 2019

/lp/de-gruyter/construction-and-stability-of-type-i-blowup-solutions-for-non-evkbZcC0WJ

- Publisher
- de Gruyter
- Copyright
- © 2019 Walter de Gruyter GmbH, Berlin/Boston
- ISSN
- 1869-6090
- eISSN
- 1869-6090
- DOI
- 10.1515/apam-2018-0168
- Publisher site
- See Article on Publisher Site

AbstractIn this note, we consider the semilinear heat system∂tu=Δu+f(v),∂tv=μΔv+g(u),μ>0,\partial_{t}u=\Delta u+f(v),\quad\partial_{t}v=\mu\Delta v+g(u),\quad\mu>0,where the nonlinearity has no gradient structure taking of the particular formf(v)=v|v|p-1 and g(u)=u|u|q-1 with p,q>1,f(v)=v\lvert v\rvert^{p-1}\quad\text{and}\quad g(u)=u\lvert u\rvert^{q-1}\quad%\text{with }p,q>1,orf(v)=epv and g(u)=equ with p,q>0.f(v)=e^{pv}\quad\text{and}\quad g(u)=e^{qu}\quad\text{with }p,q>0.We exhibit type I blowup solutions for this system and give a precise description of its blowup profiles. The method relies on a two-step procedure: the reduction of the problem to a finite-dimensional one via a spectral analysis, and then solving the finite-dimensional problem by a classical topological argument based on index theory. As a consequence of our technique, the constructed solutions are stable under a small perturbation of initial data. The results and the main arguments presented in this note can be found in our papers[T.-E. Ghoul, V. T. Nguyen and H. Zaag,Construction and stability of blowup solutions for a non-variational semilinear parabolic system,Ann. Inst. H. Poincaré Anal. Non Linéaire 35 2018, 6, 1577–1630] and[M. A. Herrero and J. J. L. Velázquez,Generic behaviour of one-dimensional blow up patterns,Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 19 1992, 3, 381–450].

Advances in Pure and Applied Mathematics – de Gruyter

**Published: ** Oct 1, 2019

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