arXiv subjects

Loth Damagui Chabi

Publications and source records attributed to Loth Damagui Chabi.

7 recordsLinked to original sources

arXiv2026

Optimal Hamilton-type gradient estimates and large time heat kernel bounds for noncompact manifolds

We derive localized and global noncompact versions of Ham\-ilton's gradient estimate for positive solutions to the heat equation on Riemannian manifolds with Ricci curvature bounded below. Our estimates are essentially optimal and significantly improve on all previous estimates of this type. As a main application, we obtain a {\it large time} logarithmic gradient estimate for the heat kernel, which is almost sharp and considerably improves on previously known results. Indeed, whereas the precise behavior was known in the small time range, the large time behavior was rather poorly understood and remained an essentially open problem, which we here solve to a large extent. As further applications, we derive a new, space only, local pseudo-Harnack inequality, as well as estimates of the spatial modulus of continuity of solutions.

math.AP↗

arXiv2026

Local existence and blow-up behavior for the diffusive Hamilton-Jacobi equation in a half-space with unbounded initial data

We study the local existence and blow-up behavior of solutions, with possible growth at space infinity, for the diffusive Hamilton-Jacobi equation $u_t-Δu=|\nabla u|^p$ ($p>1$), in a half-space with Dirichlet boundary conditions. Under optimal, polynomial growth assumptions on the initial data, characterized by the critical exponent $p/(p-1)$, we first prove local existence and uniqueness of a maximal classical solution and establish a blow-up alternative. This is complemented by a nonexistence result showing that $p/(p-1)$ is the sharp threshold for admissible growth at infinity. Next, for initial data with subcritical growth, we show global-in-time existence for $1 2$, and we derive sharp upper and lower estimates on the gradient, including a precise type~II boundary blow-up rate and a dichotomy in its behavior near the singular time. Finally, in the case $p>2$, for any type~II blow-up solution, we prove convergence after rescaling) to an explicit one-dimensional profile and providing a refined description of the asymptotic singularity formation.

math.AP↗

arXiv2026

Sharp macroscopic blow-up behavior for the parabolic-elliptic Keller-Segel system in dimensions $n\ge 3$

We study the space-time concentration or blow-up asymptotics of radially decreasing solutions of the parabolic-elliptic Keller-Segel system in the whole space or in a ball. We show that, for any solution in dimensions $3\le n\le 9$ (assuming finite mass in the whole space case), there exists a nonflat backward self-similar solution $U$ such that $$u(x,t)=(1+o(1))U(x,t),\quad\hbox{as $(x,t)\to (0,T)$.}$$ This macroscopic behavior is important from the physical point of view, since it gives a sharp description of the concentration phenomenon in the scale of the original space-time variables~$(x,t)$. It strongly improves on existing results, since such behavior was previously known (\cite{GMS}) to hold only in the microscopic scale $|x|\le O(\sqrt{T-t})$ as $t\to T$ (and in the whole space case only). As a consequence, we obtain the two-sided global estimate $$C_1\le (T-t+|x|^2)u(x,t)\le C_2\quad\hbox{in $B_R\times(T/2,T)$},$$ whose upper part only was known before (\cite{Soup-Win}), as well as the sharp final profile: $$\lim_{x\to 0} |x|^2u(x,T)=L\in(0,\infty).$$ The latter improves, with a different proof, the recent result of \cite{BZ} by excluding the possibility $L=0$. We also give extensions of these results, in higher dimensions, to type~I and to time monotone solutions. Moreover, we extend the known results on type I estimates and on convergence in similarity variables, and significantly simplify their proofs.

math.AP↗

arXiv2025

Classification of entire and ancient solutions of the diffusive Hamilton-Jacobi equation

Consider the diffusive HJ eq. with Dirichlet conditions, which arises in stochastic control as well as in KPZ type models of surface growth. It is known that, for $p>2$ and suitably large, smooth initial data, the sol. undergoes finite time gradient blowup on the boundary. On the other hand, Liouville type rigidity or classif. ppties play a central role in the study of qualitative behavior in nonlinear elliptic and parabolic problems, and notably appear in the famous BCN conjecture about one-dimensionality of solutions in a half-space. With this motivation, we study the Liouville type classif. and symmetry ppties for entire and ancient sol. in $\R^n$ and in a half-space with Dirichlet B.C. - First, we show that any ancient sol. in $\R^n$ with sublinear upper growth at infinity is necessarily constant. This result is {\it optimal}, in view of explicit examples and solves a long standing open problem. - Next we turn to the half-space problem for $p>2$ and we completely classify entire solutions: any entire sol. is stationary and one-dimensional. The assumption is sharp in view of explicit examples for $p=2$. - Then we show that the situation is also completely different for ancient sol. in a half-space: there exist nonstationary ancient sol. for all $p>1$. Nevertheless, we show that any ancient sol. is necessarily positive, and that stationarity and one-dimensionality are recovered provided a -- close to optimal -- polynomial growth restriction is imposed on the sol. - In addition we establish new and optimal, local estimates of Bernstein and Li-Yau type. The proofs of the Liouville and classif. results are delicate, based on integral estimates, a translation-compactness procedure and comparison arguments, combined with our Bernstein and Li-Yau type estimates.

math.AP↗

arXiv2025

Refined blow-up behavior for reaction-diffusion equations with non scale invariant exponential nonlinearities

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 $L(e^u)$.

math.AP↗

arXiv2025

Asymptotic blow-up behavior for the semilinear heat equation with non scale invariant nonlinearity

We characterize the asymptotic behavior near blowup points for positive solutions of the semilinear heat equation \begin{equation*} \partial_t u-Δu =f(u), \end{equation*} for nonlinearities which are genuinely non scale invariant, unlike in the standard case $f(u)=u^p$. Indeed, our results apply to a large class of nonlinearities of the form $f(u)=u^pL(u)$, where $p>1$ is Sobolev subcritical and $L$ is a slowly varying function at infinity (which includes for instance logarithms and their powers and iterates, as well as some strongly oscillating functions). More precisely, denoting by $ψ$ the unique positive solution of the corresponding ODE $y'(t)=f(y(t))$ which blows up at the same time $T$, we show that if $a\inΩ$ is a blowup point of $u$, then \begin{equation*} \lim_{t\to T}\frac{u(a+y\sqrt{T-t},t)}{ψ(t)}= 1,\quad \text{uniformly for $y$ bounded.} \end{equation*} Additional blow-up properties are obtained, including the compactness of the blow-up set for the Cauchy problem with decaying initial data.

math.AP↗

arXiv2024

Refined behavior and structural universality of the blow-up profile for the semilinear heat equation with non scale invariant nonlinearity

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.

math.AP↗