Remarks on the wave equation with a nonlinear term with respect to the velocity
Haraux, A. (1992)
Portugaliae mathematica
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Haraux, A. (1992)
Portugaliae mathematica
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Takayoshi Ogawa (2006)
Banach Center Publications
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We classify the global behavior of weak solutions of the Keller-Segel system of degenerate and nondegenerate type. For the stronger degeneracy, the weak solution exists globally in time and has a uniform time decay under some extra conditions. If the degeneracy is weaker, the solution exhibits a finite time blow up if the data is nonnegative. The situation is very similar to the semilinear case. Some additional discussion is also presented.
Esquivel-Avila, Jorge A.
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We present sufficient conditions on the initial data of an undamped Klein-Gordon equation in bounded domains with homogeneous Dirichlet boundary conditions to guarantee the blow up of weak solutions. Our methodology is extended to a class of evolution equations of second order in time. As an example, we consider a generalized Boussinesq equation. Our result is based on a careful analysis of a differential inequality. We compare our results with the ones in the literature.
E. Horst (1987)
Banach Center Publications
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Juan Luis Vázquez (2004)
Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
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We review the main mathematical questions posed in blow-up problems for reaction-diffusion equations and discuss results of the author and collaborators on the subjects of continuation of solutions after blow-up, existence of transient blow-up solutions (so-called peaking solutions) and avalanche formation as a mechanism of complete blow-up.
Balcerzak, Marek, Filipczak, Małgorzata (2018-01-08T08:56:28Z)
Acta Universitatis Lodziensis. Folia Mathematica
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Hebey, Emmanuel, Robert, Frédéric (2004)
Electronic Research Announcements of the American Mathematical Society [electronic only]
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Giga, Y., Umeda, N.
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Steve Schochet (1999)
Journées équations aux dérivées partielles
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The blow-up of solutions to a quasilinear heat equation is studied using a similarity transformation that turns the equation into a nonlocal equation whose steady solutions are stable. This allows energy methods to be used, instead of the comparison principles used previously. Among the questions discussed are the time and location of blow-up of perturbations of the steady blow-up profile.