We consider radial decreasing solutions of the semilinear heat equation with exponential nonlinearity. We provide a relatively simple proof of the sharp upper estimates for the final blowup profile and for the refined space-time behavior. We actually establish a global, upper space-time estimate, which contains those of the final and refined profiles as special cases.
Refined structures of blowup for non-collapsing maximal solution to a semilinear parabolic equation are studied. We will prove that the blowup set is empty for non-collapsing blowing-up in subcritical case, and all finite time non-collapsing blowing-up must be type II in critical case. When p>p_S=(N+2)/(N-2) for N>=3, the Hausdorff dimension of the blowup set for maximal solution whose energy is non-collapsing is shown to be no greater than N-2-4/(p-1), which answers a question proposed in [7] positively. At the end of this paper, we also present some new examples of collapsing and non-collaps…
We consider the energy critical semilinear heat equation $$ \left\{\begin{aligned} &\partial_t u-Δu =|u|^{\frac{4}{n-2}}u &\mbox{in } {\mathbb R}^n\times(0,T),\\ &u(x,0)=u_0(x), \end{aligned}\right. $$ where $ n\geq 3$, $u_0\in L^\infty({\mathbb R}^n)$, and $T\in {\mathbb R}^+$ is the first blow up time. We prove that if $ n \geq 7$ and $ u_0 \geq 0$, then any blowup must be of Type I, i.e., \[\|u(\cdot, t)\|_{L^\infty({\mathbb R}^n)}\leq C(T-t)^{-\frac{1}{p-1}}.\] A similar result holds for bounded convex domains. The proof relies on a reverse inner-outer gluing mechanism and delicate analysi…
This work studies nonnegative solutions for the Cauchy, Neumann, and Dirichlet problems of a logistic type reaction-diffusion equation. The finite time blowup results for nonnegative solutions under various restrictions on the coefficients of this equation are presented. Applying the results allows one to construct some reaction diffusion systems with the so-called diffusion-induced blowup phenomenon, particularly in the case of equal diffusion rates.
In this paper we provide an example of a class of two reaction-diffusion-ODE equations with homogeneous Neumann boundary conditions, in which Turing-type instability not only destabilizes constant steady states but also induces blow-up of nonnegative spatially heterogeneous solutions. Solutions of this problem preserve nonnegativity and uniform boundedness of the total mass. Moreover, for the corresponding system with two non-zero diffusion coefficients, all nonnegative solutions are global in time. We prove that a removal of diffusion in one of the equations leads to a finite-time blow-up of …
We consider the following parabolic system whose nonlinearity has no gradient structure: $$\left\{\begin{array}{ll} \partial_t u = Δu + e^{pv}, \quad & \partial_t v = μΔv + e^{qu}, u(\cdot, 0) = u_0, \quad & v(\cdot, 0) = v_0, \end{array}\right. \quad p, q, μ> 0, $$ in the whole space $\mathbb{R}^N$. We show the existence of a stable blowup solution and obtain a complete description of its singularity formation. The construction relies on the reduction of the problem to a finite dimensional one and a topological argument based on the index theory to conclude. In particular, our analysis uses n…
In the last twenty years, there have been significant advances in the study of the blow-up phenomenon for the critical generalized Korteweg-de Vries equation, including the determination of sufficient conditions for blowup, the stability of blowup in a refined topology and the classification of minimal mass blowup. Exotic blow-up solutions with a continuum of blow-up rates and multi-point blow-up solutions were also constructed. However, all these results, as well as numerical simulations, involve the bubbling of a solitary wave going at infinity at the blow-up time, which means that the blow-…
The existence of finite time blowup solutions for the two-dimensional Landau--Lifshitz equation is a long-standing problem, which exists in the literature at least since 2001 (E, Mathematics Unlimited--2001 and Beyond, Springer, Berlin, P.410, 2001). A more refined description in the equivariant class is given in (van den Berg and Williams, European J. Appl. Math., 24(6), 912--948, 2013). In this paper, we consider the blowup dynamics of the Landau--Lifshitz equation $$ \partial_tu=\mathfrak{a}_1u\timesΔu-\mathfrak{a}_2u\times(u\timesΔu),\quad x\in\mathbb{R}^2, $$ where $u\in\mathbb{S}^2$, $\m…
We consider the following exponential reaction-diffusion equation involving a nonlinear gradient term: $$\partial_t U = ΔU + α|\nabla U|^2 + e^U,\quad (x, t)\in\mathbb{R}^N\times[0,T), \quad α> -1.$$ We construct for this equation a solution which blows up in finite time $T > 0$ and satisfies some prescribed asymptotic behavior. We also show that the constructed solution and its gradient blow up in finite time $T$ simultaneously at the origin, and find precisely a description of its final blowup profile. It happens that the quadratic gradient term is critical in some senses, resulting in the c…
We consider reaction-diffusion equations either posed on Riemannian manifolds or in the Euclidean weighted setting, with pow\-er-type nonlinearity and slow diffusion of porous medium time. We consider the particularly delicate case $p<m$ in problem (1.1), a case largely left open in [21] even when the initial datum is smooth and compactly supported. We prove global existence for L$^m$ data, and a smoothing effect for the evolution, i.e. that solutions corresponding to such data are bounded at all positive times with a quantitative bound on their L$^\infty$ norm. As a consequence of this fact a…
We address the question whether a singularity in a three-dimensional incompressible inviscid fluid flow can occur in finite time. Analytical considerations and numerical simulations suggest high-symmetry flows being a promising candidate for a finite-time blowup. Utilizing Lagrangian and geometric non-blowup criteria, we present numerical evidence against the formation of a finite-time singularity for the high-symmetry vortex dodecapole initial condition. We use data obtained from high resolution adaptively refined numerical simulations and inject Lagrangian tracer particles to monitor geometr…
In this paper, we consider the heat flow for Yang-Mills connections on $\mathbb{R}^5 \times SO(5)$. In the $SO(5)-$equivariant setting, the Yang-Mills heat equation reduces to a single semilinear reaction-diffusion equation for which an explicit self-similar blowup solution was found by Weinkove \cite{Wei04}. We prove the nonlinear asymptotic stability of this solution under small perturbations. In particular, we show that there exists an open set of initial conditions in a suitable topology such that the corresponding solutions blow up in finite time and converge to a non-trivial self-similar…
We present a refined duality estimate for parabolic equations. This estimate entails new results for systems of reaction-diffusion equations, including smoothness and exponential convergence towards equilibrium for equations with quadratic right-hand sides in two dimensions. For general systems in any space dimension, we obtain smooth solutions of reaction-diffusion systems coming out of reversible chemistry under an assumption that the diffusion coefficients are sufficiently close one to another.
We focus on the (sharp) threshold phenomena arising in some reaction-diffusion equations supplemented with some compactly supported initial data. In the so-called ignition and bistable cases, we prove the first sharp quantitative estimate on the (sharp) threshold values. Furthermore, numerical explorations allow to conjecture some refined estimates. Last we provide related results in the case of a degenerate monostable nonlinearity "not enjoying the hair trigger effect". AMS Subject Classifications: 35K57 (Reaction-diffusion equations), 35K15 (Initial value problems for second-order parabolic …
In this paper, we mainly consider the initial boundary problem for a quasilinear parabolic equation \[ u_t-\mathrm{div}\left(|\nabla u|^{p-2}\nabla u\right)=-|u|^{β-1}u+α|u|^{q-2}u, \] where $p>1,β>0$, $q\geq1$ and $α>0$. By using Gagliardo-Nirenberg type inequality, energy method and comparison principle, the phenomena of blowup and extinction are classified completely in the different ranges of reaction exponents.
Refined blow-up criteria and global solutions for triangular cross-diffusion systems
Sorbonne Université, CNRS, Université Paris Cité, Laboratoire Jacques-Louis Lions (LJLL), Département de Mathématiques et Applications (DMA), Ecole Normale Supérieure (ENS-PSL)（索邦大学、法国国家科学研究中心、巴黎西岱大学、雅克-路易·利翁斯实验室、数学与应用系、巴黎高等师范学院）
The existence of global-in-time weak solutions to reaction-cross-diffusion systems for an arbitrary number of competing population species is proved. The equations can be derived from an on-lattice random-walk model with general transition rates. In the case of linear transition rates, it extends the two-species population model of Shigesada, Kawasaki, and Teramoto. The equations are considered in a bounded domain with homogeneous Neumann boundary conditions. The existence proof is based on a refined entropy method and a new approximation scheme. Global existence follows under a detailed balan…
We consider a bistable ($0\textless{}θ\textless{}1$ being the three constant steady states) delayed reaction diffusion equation, which serves as a model in population dynamics. The problem does not admit any comparison principle. This prevents the use of classical technics and, as a consequence, it is far from obvious to understand the behaviour of a possible travelling wave in $+\infty$. Combining refined {\it a priori} estimates and a Leray Schauder topological degree argument, we construct a travelling wave connecting 0 in $-\infty$ to \lq\lq something" which is strictly above the unstable …
