Optimal condition for blow-up of the critical $L^q$ norm for the semilinear heat equation
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arXiv
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| Format: | Preprint |
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2018
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| _version_ | 1866913810661507072 |
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| author | Mizoguchi, Noriko Souplet, Philippe |
| author_facet | Mizoguchi, Noriko Souplet, Philippe |
| contents | We shed light on a long-standing open question for the semilinear heat equation $u_t = Δu + |u|^{p-1} u$. Namely, without any restriction on the exponent $p>1$ nor on the smooth domain~$Ω$, we prove that the critical $L^q$ norm blows up whenever the solution undergoes {\it type~I~blow-up.} A~similar property is also obtained for the local critical $L^q$ norm near any blow-up point.
In view of recent results of existence of type~II blow-up solutions with bounded critical $L^q$ norm, which are counter-examples to the open question, our result seems to be essentially the best possible result in general setting. This close connection between type I blow-up and critical $L^q$ norm blow-up appears to be a completely new observation.
Our proof is rather involved and requires the combination of various ingredients. It is based on analysis in similarity variables and suitable rescaling arguments, combined with {\it backward uniqueness and unique continuation properties} for parabolic equations.
As a by-product, we obtain the nonexistence of self-similar profiles in the critical $L^q$ space. Such properties were up to now only known for $ p \le p_S $ and in radially symmetric case for $ p > p_S $, where $ p_S $ is the Sobolev exponent. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1812_11352 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | Optimal condition for blow-up of the critical $L^q$ norm for the semilinear heat equation Mizoguchi, Noriko Souplet, Philippe Analysis of PDEs 35K58, 35B40, 35B44, 35B33 We shed light on a long-standing open question for the semilinear heat equation $u_t = Δu + |u|^{p-1} u$. Namely, without any restriction on the exponent $p>1$ nor on the smooth domain~$Ω$, we prove that the critical $L^q$ norm blows up whenever the solution undergoes {\it type~I~blow-up.} A~similar property is also obtained for the local critical $L^q$ norm near any blow-up point. In view of recent results of existence of type~II blow-up solutions with bounded critical $L^q$ norm, which are counter-examples to the open question, our result seems to be essentially the best possible result in general setting. This close connection between type I blow-up and critical $L^q$ norm blow-up appears to be a completely new observation. Our proof is rather involved and requires the combination of various ingredients. It is based on analysis in similarity variables and suitable rescaling arguments, combined with {\it backward uniqueness and unique continuation properties} for parabolic equations. As a by-product, we obtain the nonexistence of self-similar profiles in the critical $L^q$ space. Such properties were up to now only known for $ p \le p_S $ and in radially symmetric case for $ p > p_S $, where $ p_S $ is the Sobolev exponent. |
| title | Optimal condition for blow-up of the critical $L^q$ norm for the semilinear heat equation |
| topic | Analysis of PDEs 35K58, 35B40, 35B44, 35B33 |
| url | https://arxiv.org/abs/1812.11352 |