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

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

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.

• Journal Article Type: Article
• Publication Date: Nov 15, 2013
• Journal: Journal of Differential Equations
• Print ISSN: 0022-0396
• 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