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Non-existence of local solutions for semilinear heat equations of Osgood type

Laister, Robert; Robinson, James C.; Sier??ga, Mikolaj

Authors

James C. Robinson

Mikolaj Sier??ga



Abstract

We establish non-existence results for the Cauchy problem of some semilinear heat equations with non-negative initial data and locally Lipschitz, non-negative source term f. Global (in time) solutions of the scalar ODE v;=f(v) exist for v(0)>0 if and only if the Osgood-type condition ∫1∞dsf(s)=∞ holds; by comparison this ensures the existence of global classical solutions of ut=δu+f(u) for bounded initial data u0∈L∞(Rn). It is natural to ask whether the Osgood condition is sufficient to ensure that the problem still admits global solutions if the initial data is in Lq(Rn) for some 1≤q0. © 2013 Elsevier Inc.

Citation

Laister, R., Robinson, J. C., & Sierżęga, M. (2013). Non-existence of local solutions for semilinear heat equations of Osgood type. Journal of Differential Equations, 255(10), 3020-3028. https://doi.org/10.1016/j.jde.2013.07.007

Journal Article Type Article
Publication Date Nov 15, 2013
Journal Journal of Differential Equations
Print ISSN 0022-0396
Electronic ISSN 1090-2732
Publisher Elsevier
Peer Reviewed Peer Reviewed
Volume 255
Issue 10
Pages 3020-3028
DOI https://doi.org/10.1016/j.jde.2013.07.007
Keywords semilinear heat equation, cauchy problem, non-existence, Osgood-type condition
Public URL https://uwe-repository.worktribe.com/output/925928
Publisher URL http://dx.doi.org/10.1016/j.jde.2013.07.007