Blow-up Patterns Research Articles

Blow-up patterns for a reaction-diffusion equation with weighted reaction in general dimension

Blow-up patterns for a reaction-diffusion equation with weighted reaction in general dimension

On singularity formation for the two-dimensional unsteady Prandtl system around the axis

We consider the two-dimensional unsteady Prandtl system. For a special class of outer Euler flows and solutions of the Prandtl system, the trace of the tangential derivative of the tangential velocity along the transversal axis solves a closed one-dimensional equation. First, we give a precise description of singular solutions for this reduced problem. A stable blow-up pattern is found, in which the blow-up point is ejected to infinity in finite time, and the solutions form a plateau with growing length. Second, in the case where, for a general analytic solution, this trace of the derivative on the axis follows the stable blow-up pattern, we show persistence of analyticity around the axis up to the blow-up time, and establish a universal lower bound of (T - t)^{7/4} for its radius of analyticity.

Self-similar blow-up patterns for a reaction-diffusion equation with weighted reaction in general dimension

<p style='text-indent:20px;'>We classify the finite time blow-up profiles for the following reaction-diffusion equation with unbounded weight:</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \partial_tu = \Delta u^m+|x|^{\sigma}u^p, $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>posed in any space dimension <inline-formula><tex-math id="M1">\begin{document}$ x\in \mathbb{R}^N $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M2">\begin{document}$ t\geq0 $\end{document}</tex-math></inline-formula> and with exponents <inline-formula><tex-math id="M3">\begin{document}$ m>1 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M4">\begin{document}$ p\in(0, 1) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M5">\begin{document}$ \sigma>2(1-p)/(m-1) $\end{document}</tex-math></inline-formula>. We prove that blow-up profiles in backward self-similar form exist for the indicated range of parameters, showing thus that the unbounded weight has a strong influence on the dynamics of the equation, merging with the nonlinear reaction in order to produce finite time blow-up. We also prove that all the blow-up profiles are <i>compactly supported</i> and might present two different types of interface behavior and three different possible <i>good behaviors</i> near the origin, with direct influence on the blow-up behavior of the solutions. We classify all these profiles with respect to these different local behaviors depending on the magnitude of <inline-formula><tex-math id="M6">\begin{document}$ \sigma $\end{document}</tex-math></inline-formula>. This paper generalizes in dimension <inline-formula><tex-math id="M7">\begin{document}$ N>1 $\end{document}</tex-math></inline-formula> previous results by the authors in dimension <inline-formula><tex-math id="M8">\begin{document}$ N = 1 $\end{document}</tex-math></inline-formula> and also includes some finer classification of the profiles for <inline-formula><tex-math id="M9">\begin{document}$ \sigma $\end{document}</tex-math></inline-formula> large that is new even in dimension <inline-formula><tex-math id="M10">\begin{document}$ N = 1 $\end{document}</tex-math></inline-formula>.</p>

Separate variable blow-up patterns for a reaction–diffusion equation with critical weighted reaction

We study the separate variable blow-up patterns associated to the following second order reaction–diffusion equation: ∂tu=Δum+|x|σum,posed for x∈RN, t≥0, where m>1, dimension N≥2 and σ>0. A new and explicit critical exponent σc=2(m−1)(N−1)3m+1is introduced and a classification of the blow-up profiles is given. The most interesting contribution of the paper is showing that existence and behavior of the blow-up patterns is split into different regimes by the critical exponent σc and also depends strongly on whether the dimension N≥4 or N∈{2,3}. These results extend previous works of the authors in dimension N=1.

Self-similar Blow-Up Profiles for a Reaction–Diffusion Equation with Critically Strong Weighted Reaction

We study the separate variable blow-up patterns associated to the following second order reaction-diffusion equation: $$ \partial_tu=\Delta u^m + |x|^{\sigma}u^m, $$ posed for $x\in\mathbb{R}^N$, $t\geq0$, where $m>1$, dimension $N\geq2$ and $\sigma>0$. A new and explicit critical exponent $$ \sigma_c=\frac{2(m-1)(N-1)}{3m+1} $$ is introduced and a classification of the blow-up profiles is given. The most interesting contribution of the paper is showing that existence and behavior of the blow-up patterns is split into different regimes by the critical exponent $\sigma_c$ and also depends strongly on whether the dimension $N\geq4$ or $N\in\{2,3\}$. These results extend previous works of the authors in dimension $N=1$.

Energy Dissipative Local Discontinuous Galerkin Methods for Keller–Segel Chemotaxis Model

In this paper, we apply the local discontinuous Galerkin (LDG) method to solve the classical Keller–Segel (KS) chemotaxis model. The exact solution of the KS chemotaxis model may exhibit blow-up patterns with certain initial conditions, and is not easy to approximate numerically. Moreover, it has been proved that there exists a definition of free energy of the KS system which dissipates with respect to time. We will construct a consistent numerical energy and prove the energy dissipation with the LDG discretization. Several numerical experiments in one and two space dimensions will be given. Especially, for solutions with blow-up (converge to Dirac delta functions), the densities of KS model are computed to be strictly positive in the numerical experiments and the energies are also numerically observed to be strictly positive and decreasing as are seen in the figures. Therefore, the scheme is stable for the KS model with blow-up solutions.

Steady States, Global Existence and Blow-up for Fourth-Order Semilinear Parabolic Equations of Cahn–Hilliard Type

Abstract Fourth-order semilinear parabolic equations of the Cahn-Hilliard-type ut + Δ2u = γu ± Δ(|u|p−1u) in Ω × ℝ+, (0.1) are considered in a smooth bounded domain Ω ⊂ ℝN with Navier-type boundary conditions on ∂Ω, or Ω = ℝN, where p > 1 and γ are given real parameters. The sign “+” in the “diffusion term” on the right-hand side means the stable case, while “−” reflects the unstable (blow-up) one, with the simplest, so called limit, canonical model for γ = 0, ut + Δ2u = ±Δ(|u|p−1u) in ℝN × R+. (0.2) The following three main problems are studied: (i) for the unstable model (0.1), with the −Δ(|u|p−1u), existence and multiplicity of classic steady states in Ω ⊂ ℝN and their global behaviour for large γ > 0; (ii) for the stable model (0.2), global existence of smooth solutions u(x, t) in ℝN × R+ for bounded initial data u0(x) in the subcritical case ; and (iii) for the unstable model (0.2), a relation between finite time blow-up and structure of regular and singular steady states in the supercritical range. In particular, three distinct families of Type I and II blow-up patterns are introduced in the unstable case.

On self-similar blow-up in evolution equations of Monge-Ampere type

We use techniques from reaction-diffusion theory to study the blow-up and existence of solutions of the parabolic Monge–Ampere (M-A) equation with power source, with the following basic 2D model $$u_{t}=-|D^{2}u|+|u|^{p-1}u\quad \mathrm{in}\quad \mathbb{R}^{2}\times \mathbb{R}_{+},$$ where in two-dimensions $|D^{2}u|=u_{xx}u_{yy}-(v_{xy})^{2}$ and p > 1 is a fixed exponent. For a class of ‘dominated concave’ and compactly supported radial initial data $u_{0}(x)\geqslant0$ , the Cauchy problem is shown to be locally well posed and to exhibit finite time blow-up that is described by similarity solutions. For p ∈ (1, 2], similarity solutions, containing domains of concavity and convexity, are shown to be compactly supported and correspond to surfaces with flat sides that persist until the blow-up time. The case p > 2 leads to single-point blow-up. Numerical computations of blow-up solutions without radial symmetry are also presented. The parabolic analogy of the parabolic M-A equation in 3D for which $|D^{2}u|$ is a cubic operator is $$u_{t}=|D^{2}u|+|u|^{p-1}u\quad \mathrm{in}\quad \mathbb{R}^{3}\times \mathbb{R}_{+},$$ and is shown to admit a wider set of (oscillatory) self-similar blow-up patterns. Regional self-similarblow-up in a cubic radial model related to the fourth-order M-A equation $$u_{t}=-|D^{4}u|+u^{3}\quad \mathrm{in}\quad \mathbb{R}^{2}\times \mathbb{R}_{+},$$ where the cubic operator $|D^{4}u|$ is the catalecticant 3 × 3 determinant is also briefly discussed.

Asymptotic behaviour, uniqueness and stability of coexistence states of a non-cooperative reaction–diffusion model of nuclear reactors

We investigate a non-cooperative reaction-diffusion model arising in the theory of nuclear reactors and are concerned with the associated steady-state problem. We determine the asymptotic behaviour of the coexistence states near the point of bifurcation from infinity, which exhibits the following very interesting spatial blow-up pattern: when the fuel temperature reaches a certain value, the free fast neutrons undergoing nuclear reaction will blow up in each spatial point of the interior of the reactor. Without any restriction on spatial dimensions, we also discuss the uniqueness and stability of the coexistence states. Our results complement and sharpen those derived in two recent works by Arioli and Lóopez-Gómez.

Numerical simulation of a kinetic model for chemotaxis

This paper is devoted to numerical simulations of a kinetic modeldescribing chemotaxis. This kinetic framework has been investigatedsince the 80's when experimental observations have shown that themotion of bacteria is due to the alternance of 'runs and tumbles'.Since parabolic and hyperbolic models do not take into account themicroscopic movement of individual cells, kinetic models have become of agreat interest. Dolak and Schmeiser (2005) have then proposed a kineticmodel describing the motion of bacteria responding to temporal gradientsof chemoattractants along their paths.An existence result for this system is provided and anumerical scheme relying on a semi-Lagrangian method is presentedand analyzed. An implementation of this scheme allows to obtainnumerical simulations of the model and observe blow-up patterns thatdiffer greatly from the case of Keller-Segel type of models.

An investigation of a nonlocal hyperbolic model for self-organization of biological groups

In this article, we introduce and study a new nonlocal hyperbolic model for the formation and movement of animal aggregations. We assume that the nonlocal attractive, repulsive, and alignment interactions between individuals can influence both the speed and the turning rates of group members. We use analytical and numerical techniques to investigate the effect of these nonlocal interactions on the long-time behavior of the patterns exhibited by the model. We establish the local existence and uniqueness and show that the nonlinear hyperbolic system does not develop shock solutions (gradient blow-up). Depending on the relative magnitudes of attraction and repulsion, we show that the solutions of the model either exist globally in time or may exhibit finite-time amplitude blow-up. We illustrate numerically the various patterns displayed by the model: dispersive aggregations, finite-size groups and blow-up patterns, the latter corresponding to aggregations which may collapse to a point. The transition from finite-size to blow-up patterns is governed by the magnitude of the social interactions and the random turning rates. The presence of these types of patterns and the absence of shocks are consequences of the biologically relevant assumptions regarding the form of the speed and the turning rate functions, as well as of the kernels describing the social interactions.

Five types of blow-up in a semilinear fourth-order reaction–diffusion equation: an analytic–numerical approach

Five types of blow-up patterns that can occur for the 4th-order semilinear parabolic equation of the reaction–diffusion type are discussed. For the semilinear heat equation ut = Δu + up, various blow-up patterns were under scrutiny since the 1980s, while the case of higher order diffusion was studied much less, regardless of the wide range of its application. The types of blow-up include the following:Type I(ss): patterns of self-similar single point blow-up, including those for which the final time profile |u(·, T−)|N(p−1)/4 is a measure;Type I(log): self-similar non-radial blow-up with angular logarithmic TW swirl;Type I(Her): non-self-similar blow-up close to stable/centre subspaces of Hermitian operators obtained via linearization about constant uniform blow-up pattern;Type II(sing): non-self-similar blow-up on stable/centre manifolds of a singular steady state in the supercritical Sobolev range p ⩾ pS = (N + 4)/(N − 4) for N > 4 andType II(LN): non-self-similar blow-up along the manifold of stationary generalized Loewner–Nirenberg type explicit solutions in the critical Sobolev case p = pS, when |u(·, T−)|N(p−1)/4 contains a measure as a singular component.There is some evidence that Type I(ss) are structurally stable (generic) patterns. However, justifying this and the existence of other proposed types of blow-up remain difficult open problems, so formal analytic and numerical methods are key in supporting some theoretical judgements.

Blow-up properties for parabolic systems with localized nonlinear source

(Communicated by S. Cui) Abstract. This paper deals with blow-up properties of solutions to a semilinear parabolic system with nonlinear localized source involved a product with local terms ut = Δu+exp{mu(x,t)+nv(x0,t)}, vt = Δv+exp{pu(x0,t)+qv(x,t)} with homogeneous Dirichlet boundary conditions. We investigate the influence of localized sources and local terms on blow-up properties for this system, and prove that: (i )w henm, q 0 this system possesses uniform blow-up profiles, in other words, the localized terms play a leading role in the blow-up profile for this case; (ii )w henm, q > 0, this system presents single point blow-up patterns, or say that local terms dominate localized terms in the blow-up profile. Moreover, the blow-up rate estimates in time and space are obtained, respectively.

Asymptotic analysis to blow-up points for the porous medium equation with a weighted non-local source

This article deals with the porous medium equation with a more complicated source term, subject to the homogeneous Dirichlet condition, where is a ball with radius R, m > 1 and the non-negative constants satisfying . We investigate how the three factors (the non-local source , the local source and the weight function a(x)) influence the asymptotic behaviour of the solutions. It is proved that (i) when p < 1, the non-local source plays a dominating role, i.e. the blow-up set of the system is the whole domain B R , a , where . (ii) When p > m, this system presents single blow-up patterns. In other words, the local term dominates the non-local term in the blow-up profile. Moreover, the blow-up rate estimate is established with more precise coefficients determined.

Hermitian Spectral Theory and Blow-Up Patterns for a Fourth-Order Semilinear Boussinesq Equation

Two families of asymptotic blow-up patterns of nonsimilarity and similarity kinds are studied in the Cauchy problem for the fourth-order semilinear wave, or Boussinesq-type, equation The first countable family is constructed by matching with linearized patterns obtained via eigenfunctions (generalized Hermite polynomials) of a related quadratic pencil of linear operators. The second family comprises nonlinear blow-up patterns given by self-similar solutions. The results have their counterparts in the classic second-order semilinear wave equation which was known to admit blow-up solutions since Keller's work in 1957.

Three types of self-similar blow-up for the fourth-order [formula omitted]-Laplacian equation with source

Self-similar blow-up behaviour for the fourth-order quasilinear p -Laplacian equation with source, u t = − ( | u x x | n u x x ) x x + | u | p − 1 u in R × R + , where n > 0 , p > 1 , is studied. Using variational setting for p = n + 1 and branching techniques for p ⁄ = n + 1 , finite and countable families of blow-up patterns of the self-similar form u S ( x , t ) = ( T − t ) − 1 p − 1 f ( y ) , where y = x / ( T − t ) β , β = − p − ( n + 1 ) 2 ( n + 2 ) ( p − 1 ) , are described by an analytic-numerical approach. Three parameter ranges: p = n + 1 (regional), p > n + 1 (single point), and 1 < p < n + 1 (global blow-up) are studied. This blow-up model is motivated by the second-order reaction-diffusion counterpart u t = ( | u x | n u x ) x + u p ( u ≥ 0 ) that was studied in the middle of the 1980s, while first results on blow-up of solutions were established by Tsutsumi in 1972.

Blow-up similarity solutions of the fourth-order unstable thin film equation

We study blow-up behaviour of solutions of the fourth-order thin film equationwhich contains a backward (unstable) diffusion term. Our main goal is a detailed study of the case of the first critical exponentwhereN≥ 1 is the space dimension. We show that the free-boundary problem with zero contact angle and zero-flux conditions admits continuous sets (branches) of blow-up self-similar solutions. For the Cauchy problem inRN×R+, we detect compactly supported blow-up patterns, which have infinitely many oscillations near interfaces and exhibit a “maximal” regularity there. As a key principle, we use the fact that, for small positiven, such solutions are close to the similarity solutions of the semilinear unstable limit Cahn-Hilliard equationwhich are better understood and have been studied earlier [19]. We also discuss some general aspects of formation of self-similar blow-up singularities for other values ofp.

Localization of blow-up points for a nonlinear nonlocal porous medium equation

This paper deals with the porous medium equation with a nonlinearnonlocal source$u_t=\Delta u^m + au^p\int_\Omega u^q dx,\quad x\in \Omega, t>0$subject to homogeneous Dirichlet condition. We investigate theinfluence of the nonlocal source and local term on blow-upproperties for this system. It is proved that: (i) when $p\leq 1$,the nonlocal source plays a dominating role, i.e. the system hasglobal blow-up and the blow-up profile which is uniformly away fromthe boundary either at polynomial scale or logarithmic scale isobtained. (ii) When $p> m$, this system presents single blow-up pattern. In other words,the local term dominates the nonlocal term in the blow-up profile.This extends the work of Li and Xie in Appl. Math. Letter, 16(2003) 185--192.

A further blow-up analysis for a localized porous medium equation

This paper deals with the porous medium equation with a localized source v τ = Δ v m + av p 1 v q 1 ( x 0 , τ ) , x ∈ Ω , τ > 0 subject to homogeneous Dirichlet condition. We investigate the influence of the localized source and local term on blow-up properties for this system. It is proved that: (i) when p 1 ⩽ 1, the localized source plays a dominating role, i.e. the system has global blow-up and the uniformly blow-up profile is obtained. (ii) When p 1 > 1, we obtain the blow-up rate estimates under some appropriate hypotheses on initial datum. Moreover, if p 1 > m, this system presents single blow-up pattern. In other words, the local term dominates the localized term in the blow-up profile. This extends and generalizes a recent work of Chen and Xie [Y. Chen, C. Xie, Blow-up for a porous medium equation with a localized source, Appl. Math. and Comput., 159 (2004) 79–93], which only considered the blow-up profile in the special case p 1 = 0.

GLOBAL AND NON-GLOBAL SOLUTIONS OF A NONLINEAR PARABOLIC EQUATION

We study the global and non-global existence of positive solutions of a nonlinear parabolic equation. For this, we consider the forward and backward self-similar solutions of this equation. We obtain a family of radial symmetric global solutions which tend to zero as the time tends infinity. Next, we show that there are initial data for which the corresponding solutions blow up in finite time. Finally, we also construct some self-similar single-point blow-up patterns with different oscillations.