Refined blow-up behavior for reaction-diffusion equations with non scale invariant exponential nonlinearities
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866909663941885952 |
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| author | Chabi, Loth Damagui |
| author_facet | Chabi, Loth Damagui |
| contents | We consider positive radial decreasing blow-up solutions of the semilinear heat equation \begin{equation*}
u_t-Δu=f(u):=e^{u}L(e^{u}),\quad x\in Ω,\ t>0,
\end{equation*}
where $Ω=\mathbb{R}^n$ or $Ω=B_R$ and $L$ is a slowly varying function
(which includes for instance logarithms and their powers and iterates, as well as some strongly oscillating unbounded functions).
We characterize the aymptotic blow-up behavior and obtain the sharp, global blow-up profile in the scale of the original variables $(x, t)$. Namely,
assuming for instance $u_t\ge 0$, we have \begin{equation*}
u(x,t)=G^{-1}\bigg(T-t+\frac{1}{8}\frac{|x|^2}{|\log |x||}\bigg)+o(1)\quad \ \hbox{as $(x,t)\to (0,T)$, where } \quad G(X)=\int_{X}^{\infty} \frac{ds}{f(s)}ds.
\end{equation*}
This estimate in particular provides the sharp final space profile and the refined space-time profile. For exponentially growing nonlinearities, such results were up to now available only in the scale invariant case $f(u)=e^u$. Moreover, this displays a universal structure of the global blow-up profile, given by the resolvent $G^{-1}$ of the ODE composed with a fixed time-space building block, which is robust with respect to the factor $L(e^u)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_06426 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Refined blow-up behavior for reaction-diffusion equations with non scale invariant exponential nonlinearities Chabi, Loth Damagui Analysis of PDEs 35K58, 35B44, 35B40 We consider positive radial decreasing blow-up solutions of the semilinear heat equation \begin{equation*} u_t-Δu=f(u):=e^{u}L(e^{u}),\quad x\in Ω,\ t>0, \end{equation*} where $Ω=\mathbb{R}^n$ or $Ω=B_R$ and $L$ is a slowly varying function (which includes for instance logarithms and their powers and iterates, as well as some strongly oscillating unbounded functions). We characterize the aymptotic blow-up behavior and obtain the sharp, global blow-up profile in the scale of the original variables $(x, t)$. Namely, assuming for instance $u_t\ge 0$, we have \begin{equation*} u(x,t)=G^{-1}\bigg(T-t+\frac{1}{8}\frac{|x|^2}{|\log |x||}\bigg)+o(1)\quad \ \hbox{as $(x,t)\to (0,T)$, where } \quad G(X)=\int_{X}^{\infty} \frac{ds}{f(s)}ds. \end{equation*} This estimate in particular provides the sharp final space profile and the refined space-time profile. For exponentially growing nonlinearities, such results were up to now available only in the scale invariant case $f(u)=e^u$. Moreover, this displays a universal structure of the global blow-up profile, given by the resolvent $G^{-1}$ of the ODE composed with a fixed time-space building block, which is robust with respect to the factor $L(e^u)$. |
| title | Refined blow-up behavior for reaction-diffusion equations with non scale invariant exponential nonlinearities |
| topic | Analysis of PDEs 35K58, 35B44, 35B40 |
| url | https://arxiv.org/abs/2502.06426 |