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Var."],"published-print":{"date-parts":[[2023,5]]},"abstract":"Abstract<\/jats:title>We give a sufficient condition for non-existence of global nonnegative mild solutions of the Cauchy problem for the semilinear heat equation $$u' = Lu + f(u)$$<\/jats:tex-math>\n \n \n u<\/mml:mi>\n \u2032<\/mml:mo>\n <\/mml:msup>\n =<\/mml:mo>\n L<\/mml:mi>\n u<\/mml:mi>\n +<\/mml:mo>\n f<\/mml:mi>\n \n (<\/mml:mo>\n u<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> in $$L^p(X, m)$$<\/jats:tex-math>\n \n \n L<\/mml:mi>\n p<\/mml:mi>\n <\/mml:msup>\n \n (<\/mml:mo>\n X<\/mml:mi>\n ,<\/mml:mo>\n m<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> for $$p \\in [1,\\infty )$$<\/jats:tex-math>\n \n p<\/mml:mi>\n \u2208<\/mml:mo>\n [<\/mml:mo>\n 1<\/mml:mn>\n ,<\/mml:mo>\n \u221e<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, where (X<\/jats:italic>,\u00a0m<\/jats:italic>) is a $$\\sigma $$<\/jats:tex-math>\n \u03c3<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>-finite measure space, L<\/jats:italic> is the infinitesimal generator of a sub-Markovian strongly continuous semigroup of bounded linear operators in $$L^p(X, m)$$<\/jats:tex-math>\n \n \n L<\/mml:mi>\n p<\/mml:mi>\n <\/mml:msup>\n \n (<\/mml:mo>\n X<\/mml:mi>\n ,<\/mml:mo>\n m<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, and f<\/jats:italic> is a strictly increasing, convex, continuous function on $$[0,\\infty )$$<\/jats:tex-math>\n \n [<\/mml:mo>\n 0<\/mml:mn>\n ,<\/mml:mo>\n \u221e<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> with $$f(0) = 0$$<\/jats:tex-math>\n \n f<\/mml:mi>\n (<\/mml:mo>\n 0<\/mml:mn>\n )<\/mml:mo>\n =<\/mml:mo>\n 0<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and $$\\int _1^\\infty 1\/f < \\infty $$<\/jats:tex-math>\n \n \n \u222b<\/mml:mo>\n 1<\/mml:mn>\n \u221e<\/mml:mi>\n <\/mml:msubsup>\n 1<\/mml:mn>\n \/<\/mml:mo>\n f<\/mml:mi>\n <<\/mml:mo>\n \u221e<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. Since we make no further assumptions on the behaviour of the diffusion, our main result can be seen as being about the competition between the diffusion represented by L<\/jats:italic> and the reaction represented by f<\/jats:italic> in a general setting. We apply our result to Laplacians on manifolds, graphs, and, more generally, metric measure spaces with a heat kernel. In the process, we recover and extend some older as well as recent results in a unified framework.\n<\/jats:p>","DOI":"10.1007\/s00526-023-02482-x","type":"journal-article","created":{"date-parts":[[2023,4,25]],"date-time":"2023-04-25T08:02:28Z","timestamp":1682409748000},"update-policy":"http:\/\/dx.doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Blow-up of nonnegative solutions of an abstract semilinear heat equation with convex source"],"prefix":"10.1007","volume":"62","author":[{"given":"Daniel","family":"Lenz","sequence":"first","affiliation":[]},{"ORCID":"http:\/\/orcid.org\/0000-0002-7918-0715","authenticated-orcid":false,"given":"Marcel","family":"Schmidt","sequence":"additional","affiliation":[]},{"given":"Ian","family":"Zimmermann","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2023,4,25]]},"reference":[{"key":"2482_CR1","doi-asserted-by":"crossref","unstructured":"Barlow, M.T.: Diffusions on fractals. 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