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 …
This paper concerns with the explicit blowup phenomenon for 3D incompressible MHD equations in R^3. More precisely, we find two family of explicit blowup solutions for 3D incompressible MHD equations in R^3. One family of solutions admit the smooth initial data, and the initial data of another family of solutions are not smooth. The energy of those solutions is infinite. Moreover, our results tell us that the blowup phenomenon of 3D incompressible MHD can only take place in the velocity field of the fluid, but no blowup for the magnetic field.
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…
This paper is concerned with the blowup phenomenon of stochastic parabolic equations both on bounded domain and in the whole space. We introduce a new method to study the blowup phenomenon on bounded domain. Comparing with the existing results, we delete the assumption that the solutions to stochastic heat equations are non-negative. Then the blowup phenomenon in the whole space is obtained by using the properties of heat kernel. We obtain that the solutions will blow up in finite time for nontrivial initial data.
In this article we study bubbling solutions of regular $SU(3)$ Toda systems defined on a Riemann surface. There are two major difficulties corresponding to the profile of bubbling solutions: partial blowup phenomenon and bubble accumulation. We prove that when both parameters tend to critical positions, if there is one fully bubbling blowup point, then under one curvature assumption, all the blowup solutions near a blowup point satisfy a spherical Harnack inequality, which completely rules out the bubble-accumulation phenomenon. This fact is crucial for a number of applications.
We study the blowup behavior of a class of strongly perturbed wave equations with a focusing supercritical power nonlinearity in three spatial dimensions. We show that the ODE blowup profile of the unperturbed equation still describes the asymptotics of stable blowup. As a consequence, stable ODE-type blowup is seen to be a universal phenomenon that exists in a large class of semilinear wave equations.
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…
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-…
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 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…
For the critical focusing wave equation $\Box u = u^5$ on $\R^{3+1}$ in the radial case, we establish the role of the "center stable" manifold $Σ$ constructed in \cite{KS} near the ground state $(W,0)$ as a threshold between type I blowup and scattering to zero, establishing a conjecture going back to numerical work by Bizoń, Chmaj, Tabor. The underlying topology is stronger than the energy norm.
It is known that smooth solutions to the non-isentropic Navier-Stokes equations without heat-conductivity may lose their regularities in finite time in the presence of vacuum. However, in spite of the recent progress on such blowup phenomenon, it remain to give a possible blowup mechanism. In this paper, we present a simple continuation principle for such system, which asserts that the concentration of the density or the temperature occurs in finite time for a large class of smooth initial data, which is responsible for the breakdown of classical solutions. It also give an affirmative answer t…
The theory of linear dispersive equations predicts that waves should spread out and disperse over time. However, it is a remarkable phenomenon, observed both in theory and practice, that once nonlinear effects are taken into account, \emph{solitary wave} or \emph{soliton} solutions can be created, which can be stable enough to persist indefinitely. The construction of such solutions is relatively straightforward, but the fact that they are \emph{stable} requires some significant amounts of analysis to establish, in part due to symmetries in the equation (such as translation invariance) which c…
In this paper, we consider the following semi-linear complex heat equation \begin{eqnarray*} \partial_t u = Δu + u^p, u \in \mathbb{C} \end{eqnarray*} in $\mathbb{R}^n,$ with an arbitrary power $p,$ $ p > 1$. In particular, $p$ can be non integer and even irrational. We construct for this equation a complex solution $u = u_1 + i u_2$, which blows up in finite time $T$ and only at one blowup point $a.$ Moreover, we also describe the asymptotics of the solution by the following final profiles: \begin{eqnarray*} u(x,T) &\sim & \left[ \frac{(p-1)^2 |x-a|^2}{ 8 p |\ln|x-a||}\right]^{-\frac{1}{p-1}}…
In this paper, we are interested in the mathematical model of MEMS devices which is presented by the following equation on $(0,T) \times Ω:$ \begin{eqnarray*} \partial_t u = Δu +\displaystyle \frac{λ}{ (1-u)^2 \left( 1 +\displaystyle γ\int_Ω \frac{1}{1-u} dx \right)^2}, \quad 0 \leq u <1, \end{eqnarray*} where $Ω$ is a bounded domain in $\mathbb{R}^n$ and $λ, γ> 0$. In this work, we have succeeded to construct a solution which quenches in finite time T only at one interior point $a \in Ω$. In particular, we give a description of the quenching behavior according to the following final profile $…
In this paper, we consider concentration phenomenon of semiclassical states to the following $2M$-component reaction-diffusion system in $\R \times \R^N$, \begin{align*} \left\{ \begin{aligned} \partial_t u &=\eps^2 Δ_x u-u-V(x)v + \partial_v H(u, v),\\ \partial_t v &=-\eps^2 Δ_x v+v + V(x)u - \partial_u H(u, v), \end{aligned} \right. \end{align*} where $M \geq 1$, $N \geq 1$, $\eps>0$ is a small parameter, $V \in C^1(\R^N, \, \R)$, $H \in C^1(\R^M \times \R^M, \, \R)$ and $(u, v): \R \times \R^N \to \R^M \times \R^M$. It is proved that there exist semiclassical states concentrating around the…
Let $Ω$ be a flat torus and $G$ be the green's function of $-Δ$ on $Ω$. One intriguing mystery of $G$ is how the number of its critical points is related to blowup solutions of certain PDEs. In this article we prove that for the following equation that describes a Chern-Simons model in Gauge theory: \begin{equation}\label{e103} \left\{ \begin{array}{ll} Δu_1+\frac{1}{\varepsilon^2}e^{u_2}(1-e^{u_1})=8πδ_{p_{1}} Δu_2+\frac{1}{\varepsilon^2}e^{u_1}(1-e^{u_2})=8πδ_{p_{2}} \end{array} \text{ in }\quad Ω\right., \quad p_1-p_2 \mbox{ is a half period}, \end{equation} if fully bubbling solutions of L…
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.
We consider absorbing chemical reactions in a fluid flow modeled by the coupled advection-reaction-diffusion equations. In these systems, the interplay between chemical diffusion and fluid transportation causes the enhanced dissipation phenomenon. We show that the enhanced dissipation time scale, together with the reaction coupling strength, determines the characteristic time scale of the reaction.
In the mathematical physics literature, there are heuristic arguments, going back three decades, suggesting that for an open set of initially smooth solutions to the Einstein-vacuum equations in high dimensions, stable, approximately monotonic curvature singularities can dynamically form along a spacelike hypersurface. In this article, we study the Cauchy problem and give a rigorous proof of this phenomenon in sufficiently high dimensions, thereby providing the first constructive proof of stable curvature blowup (without symmetry assumptions) along a spacelike hypersurface as an effect of pure…
We consider the diffusive Hamilton-Jacobi equation $$u_t-Δu=|\nabla u|^p,$$ with Dirichlet boundary conditions in two space dimensions, which arises in the KPZ model of growing interfaces. For $p>2$, solutions may develop gradient singularities on the boundary in finite time, and examples of single-point gradient blowup on the boundary are known, but the space-profile in the tangential direction has remained a completely open problem. In the parameter range $2<p\le 3$, for the case of a flat boundary and an isolated singularity at the origin, we give an answer to this question, obtaining the p…
We address the question: Why may reaction-diffusion equations with hysteretic nonlinearities become ill-posed and how to amend this? To do so, we discretize the spatial variable and obtain a lattice dynamical system with a hysteretic nonlinearity. We analyze a new mechanism that leads to appearance of a spatio-temporal pattern called {\it rattling}: the solution exhibits a propagation phenomenon different from the classical traveling wave, while the hysteretic nonlinearity, loosely speaking, takes a different value at every second spatial point, independently of the grid size. Such a dynamics …
This paper studies the phenomenon of invasion for heterogeneous reaction-diffusion equations in periodic domains with monostable and combustion reaction terms. We give an answer to a question rised by Berestycki, Hamel and Nadirashvili in [5] concerning the connection between the speed of invasion and the speed of fronts. To do so, we extend the classical Freidlin-Gartner formula to such equations, using a geometrical argument devised by Rossi in [17], and derive some bounds on the speed of fronts using estimates on the heat kernel.
In this paper, we provide a mathematical framework in studying the wave propagation with the annihilation phenomenon in excitable media. We deal with the existence and uniqueness of solutions to a one-dimensional free boundary problem (called a reaction--interface system) arising from the singular limit of a FitzHugh--Nagumo type reaction--diffusion system. Because of the presence of the annihilation, interfaces may intersect each other. We introduce the notion of weak solutions to study the continuation of solutions beyond the annihilation time. Under suitable conditions, we show that the fre…
It is well known from the work of [2] that the Fujita phenomenon for reaction-diffusion evolution equations with power nonlinearities does not occur on the hyperbolic space $\mathbb{H}^N$, thus marking a striking difference with the Euclidean situation. We show that, on classes of manifolds in which the bottom $Λ$ of the $L^2$ spectrum of $-Δ$ is strictly positive (the hyperbolic space being thus included), a different version of the Fujita phenomenon occurs for other kinds of nonlinearities, in which the role of the critical Fujita exponent in the Euclidean case is taken by $Λ$. Such nonlinea…
We study the velocity of travelling waves of a reaction-diffusion system coupling a standard reaction-diffusion equation in a strip with a one-dimensional diffusion equation on a line. We show that it grows like the square root of the diffusivity on the line. This generalises a result of Berestycki, Roquejoffre and Rossi in the context of Fisher-KPP propagation where the question could be reduced to algebraic computations. Thus, our work shows that this phenomenon is a robust one. The ratio between the asymptotic velocity and the square root of the diffusivity on the line is characterised as t…
We focus on the spreading properties of solutions of monostable reaction-diffusion equations. Initial data are assumed to have heavy tails, which tends to accelerate the invasion phenomenon. On the other hand, the nonlinearity involves a weak Allee effect, which tends to slow down the process. We study the balance between the two effects. For algebraic tails, we prove the exact separation between "no acceleration and acceleration". This implies in particular that, for tails exponentially unbounded but lighter than algebraic , acceleration never occurs in presence of an Allee effect. This is in…
We consider one-dimensional reaction-diffusion equations for a large class of spatially periodic nonlinearities (including multistable ones) and study the asymptotic behavior of solutions with Heaviside type initial data. Our analysis reveals some new dynamics where the profile of the propagation is not characterized by a single front, but by a layer of several fronts which we call a terrace. Existence and convergence to such a terrace is proven by using an intersection number argument, without much relying on standard linear analysis. Hence, on top of the peculiar phenomenon of propagation th…
We explore a mechanism of pattern formation arising in processes described by a system of a single reaction-diffusion equation coupled with ordinary differential equations. Such systems of equations arise from the modeling of interactions between cellular processes and diffusing growth factors. We focused on the model of early carcinogenesis proposed by Marciniak-Czochra and Kimmel, which is an example of a wider class of pattern formation models with an autocatalytic non-diffusing component. We present a numerical study showing emergence of periodic and irregular spike patterns due to diffusi…
We consider a multi-species reaction-diffusion system that arises in epidemiology to describe the spread of several strains, or variants, of a disease in a population. Our model is a natural spatial, multi-species, extension of the classical SIR model of Kermack and McKendrick. First, we study the long-time behavior of the solutions and show that there is a "selection via propagation" phenomenon: starting with N strains, only a subset of them - that we identify - propagates and invades space, with some given speeds that we compute. Then, we obtain some qualitative properties concerning the eff…
We study long time dynamics of combustive processes in random media, modeled by reaction-diffusion equations with random ignition reactions. One expects that under reasonable hypotheses on the randomness, large space-time scale dynamics of solutions to these equations is almost surely governed by a different effective PDE, which should be a homogeneous Hamilton-Jacobi equation. While this was previously proved in one dimension as well as for isotropic reactions in several dimensions (i.e., with radially symmetric laws), we provide here the first proof of this phenomenon in the general non-isot…
The aim of this paper is to contribute to the understanding of the pattern formation phenomenon in reaction-diffusion equations coupled with ordinary differential equations. Such systems of equations arise, for example, from modeling of interactions between cellular processes such as cell growth, differentiation or transformation and diffusing signaling factors. We focus on stability analysis of solutions of a prototype model consisting of a single reaction-diffusion equation coupled to an ordinary differential equation. We show that such systems are very different from classical reaction-diff…
