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Construction and stability of type I blowup solutions for non-variational semilinear parabolic systems

Construction and stability of type I blowup solutions for non-variational semilinear parabolic... AbstractIn this note, we consider the semilinear heat system∂t⁡u=Δ⁢u+f⁢(v),∂t⁡v=μ⁢Δ⁢v+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)=ep⁢v and g⁢(u)=eq⁢u 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]. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Advances in Pure and Applied Mathematics de Gruyter

Construction and stability of type I blowup solutions for non-variational semilinear parabolic systems

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Publisher
de Gruyter
Copyright
© 2019 Walter de Gruyter GmbH, Berlin/Boston
ISSN
1869-6090
eISSN
1869-6090
DOI
10.1515/apam-2018-0168
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Abstract

AbstractIn this note, we consider the semilinear heat system∂t⁡u=Δ⁢u+f⁢(v),∂t⁡v=μ⁢Δ⁢v+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)=ep⁢v and g⁢(u)=eq⁢u 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].

Journal

Advances in Pure and Applied Mathematics – de Gruyter

Published: Oct 1, 2019

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