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Non-existence of local solutions for semilinear heat equations of Osgood type
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Laister, Robert, Robinson, James C. and Sierżęga, Mikolaj (2013) Non-existence of local solutions for semilinear heat equations of Osgood type. Journal of Differential Equations, Volume 255 (Number 10). pp. 3020-3028. doi:10.1016/j.jde.2013.07.007 ISSN 0022-0396.
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Official URL: http://dx.doi.org/10.1016/j.jde.2013.07.007
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 View the MathML source exist for v(0)>0 if and only if the Osgood-type condition View the MathML source 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⩽q<∞. Here we answer this question in the negative, and in fact show that there are initial conditions for which there exists no local solution in View the MathML source for t>0.
Item Type: | Journal Article | ||||
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Subjects: | Q Science > QA Mathematics | ||||
Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||
Library of Congress Subject Headings (LCSH): | Heat equations, Cauchy problem | ||||
Journal or Publication Title: | Journal of Differential Equations | ||||
Publisher: | Academic Press | ||||
ISSN: | 0022-0396 | ||||
Official Date: | 2013 | ||||
Dates: |
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Volume: | Volume 255 | ||||
Number: | Number 10 | ||||
Page Range: | pp. 3020-3028 | ||||
DOI: | 10.1016/j.jde.2013.07.007 | ||||
Status: | Peer Reviewed | ||||
Publication Status: | Published | ||||
Access rights to Published version: | Restricted or Subscription Access | ||||
Funder: | Engineering and Physical Sciences Research Council (EPSRC) | ||||
Grant number: | EP/I019138/1, EP/G007470/1, EP/P50578X/1 (EPSRC) |
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