Refined behavior and structural universality of the blow-up profile for the semilinear heat equation with non scale invariant nonlinearity
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| author | Chabi, Loth Damagui Souplet, Philippe |
| author_facet | Chabi, Loth Damagui Souplet, Philippe |
| contents | We consider the semilinear heat equation $$u_t-Δu=f(u) $$
for a large class of non scale invariant nonlinearities of the form
$f(u)=u^pL(u)$, where $p>1$ is Sobolev subcritical and $L$ is a slowly varying function (which includes for instance logarithms and their powers and iterates, as well as some strongly oscillating functions).
For any positive radial decreasing blow-up solution, we obtain the sharp, global blow-up profile in the scale of the original variables $(x, t)$, which takes the form:
$$u(x,t)=(1+o(1))\,G^{-1}\bigg(T-t+\frac{p-1}{8p}\frac{|x|^2}{|\log |x||}\bigg),
\ \hbox{as $(x,t)\to (0,T)$, \quad where } G(X)=\int_{X}^{\infty}\frac{ ds}{f(s)}.$$ This estimate in particular provides the sharp final space profile and the refined space-time profile.
As a remarkable fact and completely new observation, our results reveal a {\it structural universality} of the global blow-up profile, being given by the "resolvent" $G^{-1}$ of the ODE, composed with a universal, time-space building block, which is the same as in the pure power case. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_11863 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Refined behavior and structural universality of the blow-up profile for the semilinear heat equation with non scale invariant nonlinearity Chabi, Loth Damagui Souplet, Philippe Analysis of PDEs 35K58, 35B44, 35B40 We consider the semilinear heat equation $$u_t-Δu=f(u) $$ for a large class of non scale invariant nonlinearities of the form $f(u)=u^pL(u)$, where $p>1$ is Sobolev subcritical and $L$ is a slowly varying function (which includes for instance logarithms and their powers and iterates, as well as some strongly oscillating functions). For any positive radial decreasing blow-up solution, we obtain the sharp, global blow-up profile in the scale of the original variables $(x, t)$, which takes the form: $$u(x,t)=(1+o(1))\,G^{-1}\bigg(T-t+\frac{p-1}{8p}\frac{|x|^2}{|\log |x||}\bigg), \ \hbox{as $(x,t)\to (0,T)$, \quad where } G(X)=\int_{X}^{\infty}\frac{ ds}{f(s)}.$$ This estimate in particular provides the sharp final space profile and the refined space-time profile. As a remarkable fact and completely new observation, our results reveal a {\it structural universality} of the global blow-up profile, being given by the "resolvent" $G^{-1}$ of the ODE, composed with a universal, time-space building block, which is the same as in the pure power case. |
| title | Refined behavior and structural universality of the blow-up profile for the semilinear heat equation with non scale invariant nonlinearity |
| topic | Analysis of PDEs 35K58, 35B44, 35B40 |
| url | https://arxiv.org/abs/2404.11863 |