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

Tej-eddine Ghoul, Van Tien Nguyen, Hatem Zaag

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Abstract

In this note, we consider the semilinear heat system = Δ = μ Δ (u), μ > 0 =, where the nonlinearity has no gradient structure taking of the particular form f (v) = v | v | p - 1 and (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, or f (v) = e p v and g (u) = e q u with p, q > 0. f(v)=e^{pv}\quad\text{and}\quad g(u)= {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].

Original languageEnglish (US)
JournalAdvances in Pure and Applied Mathematics
DOIs
StatePublished - Jan 1 2019

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Semilinear Parabolic Systems
Blow-up Solution
Blow-up
Index Theory
Blow-up of Solutions
Spectral Analysis
Small Perturbations
Semilinear
Heat
Nonlinearity
Gradient
Norm
Text

Keywords

  • blowup profile
  • Blowup solution
  • semilinear parabolic system
  • stability

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

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title = "Construction and stability of type i blowup solutions for non-variational semilinear parabolic systems",
abstract = "In this note, we consider the semilinear heat system = Δ = μ Δ (u), μ > 0 =, where the nonlinearity has no gradient structure taking of the particular form f (v) = v | v | p - 1 and (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, or f (v) = e p v and g (u) = e q u with p, q > 0. f(v)=e^{pv}\quad\text{and}\quad g(u)= {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. Poincar{\'e} Anal. Non Lin{\'e}aire 35 2018, 6, 1577-1630] and [M. A. Herrero and J. J. L. Vel{\'a}zquez, Generic behaviour of one-dimensional blow up patterns, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 19 1992, 3, 381-450].",
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AU - Ghoul, Tej-eddine

AU - Nguyen, Van Tien

AU - Zaag, Hatem

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N2 - In this note, we consider the semilinear heat system = Δ = μ Δ (u), μ > 0 =, where the nonlinearity has no gradient structure taking of the particular form f (v) = v | v | p - 1 and (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, or f (v) = e p v and g (u) = e q u with p, q > 0. f(v)=e^{pv}\quad\text{and}\quad g(u)= {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].

AB - In this note, we consider the semilinear heat system = Δ = μ Δ (u), μ > 0 =, where the nonlinearity has no gradient structure taking of the particular form f (v) = v | v | p - 1 and (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, or f (v) = e p v and g (u) = e q u with p, q > 0. f(v)=e^{pv}\quad\text{and}\quad g(u)= {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].

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