Backward Self-similar Solutions Research Articles

Self-similarity in chemotaxis systems

We consider a system which describes the scaling limit of several chemotaxis systems. We focus on self-similarity, and review some recent results on forward and backward self-similar solutions to the system.

Self-similar solutions to the isothermal compressible Navier–Stokes equations

We investigate the self-similar solutions to the isothermal compressible Navier-Stokes equations. The aim of this paper is to show that there exist neither forward nor backward self-similar solutions with finite total energy. This generalizes the results for the incompressible case in Necas, J., Ruzicka, M. & Sverak, V. (1996, On Leray's self-similar solutions of the Navier-Stokes equations. Acta. Math., 176, 283-294), and is consistent with the (unproved) existence of regular solutions globally in time for the compressible Navier-Stokes equations.

Existence of type II blowup solutions for a semilinear heat equation with critical nonlinearity

We consider the blowup rate of solutions for a semilinear heat equation u t = Δ u + | u | p − 1 u , x ∈ Ω ⊂ R N , t > 0 , with critical power nonlinearity p = ( N + 2 ) / ( N − 2 ) and N ⩾ 3 . First we investigate the profiles of backward self-similar solutions by making use of the variational method, and then, by employing the intersection comparison argument with a particular self-similar solution, we derive the criteria of the blowup rate of solutions, assuming the positivity of solutions in backward space–time parabola. In particular, we show the existence of the so-called type II blowup solutions for the Cauchy–Dirichlet problems on suitable shrinking domains in the case N = 3 .

Type II blowup of solutions to a simplified Keller–Segel system in two dimensional domains

We consider a parabolic–elliptic system which is introduced as a simplified version of the so-called Keller–Segel system. In particular, we consider the system in a bounded domain of two dimensional Euclidean space. In that situation, we can find solutions to the system blowing up in finite time. Then, these solutions become the sum of an L 1 -function and delta functions at the blowup time. As regards the blowup speed, there exists a radial solution whose blowup speed is faster than that of backward self-similar solutions. We refer to that blowup as Type II blowup. In this paper, we investigate whether finite time blowup solutions to the system exhibit Type II blowup or not.

A fast blowup solution to an elliptic-parabolic system related to chemotaxis

We consider radial blowup solutions to an elliptic-parabolic system in $N$-dimensional Euclidean space. The system is introduced to describe several phenomena, for example, motion of bacteria by chemotaxis and equilibrium of self-attracting clusters. In the case where $N \geq 3$, we can find positive and radial backward self-similar solutions which blow up in finite time. In the present paper, in the case where $N \geq 11$, we show the existence of a radial blowup solution whose blowup speed is faster than the one of backward self-similar solutions, by using so-called asymptotic matched expansion techniques.

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.

Self-similar solutions for the kinematic model equation of spiral waves

We present a class of self-similar solutions of the kinematic model equation, introduced by V.A. Davydov, A.S. Mikhailov, and V.S. Zykov. This equation is designed to describe the dynamics of spiral waves in excitable media. In this model, the sharply located spiral fronts are regarded as planar curves. If the tip neither grows nor retracts in the tangential direction and if their normal velocity (with the eikonal approximation) is assumed to possess no driving force, then the kinematic equation admits self-similar solutions with nonzero curvature. We show the global structure of both forward and backward self-similar solutions, which implies mathematically the existence of various types of spiral waves.

On the energy spectrum for weak solutions of the Navier–Stokes equations

We consider the decay at high wavenumbers of the energy spectrum for weak solutions to the three-dimensional forced Navier–Stokes equation in the whole space. We observe that known regularity criteria imply that solutions are regular if the energy density decays at a sufficiently fast rate. This result applies also to a class of solutions with infinite global energy by localizing the Navier–Stokes equation. We consider certain modified Leray backward self-similar solutions, which belong to this class, and show that their energy spectrum decays at the critical rate for regularity. Therefore, this rate of decay is consistent with the appearance of an isolated self-similar singularity.

Single-point blow-up patterns for a nonlinear parabolic equation

We study a nonlinear parabolic equation with a superlinear reaction term. By studying the backward self-similar solutions for this equation, we construct a finite number of self-similar single-point blow-up patterns with different oscillations.

Self-Similar Blow-Up and Hausdorff Dimension Estimates for a Class of Parabolic Free Boundary Problems

We consider variational solutions of the quenching problem $\partial_t u - \Delta u = - u^\gamma \chi_{\{u>0\}}$ with exponent $\gamma \in (-{1\over 3},0)$ and solutions of the heat equation with Bernoulli-type condition on the free boundary which arises in combustion theory, $\partial_t v - \Delta v =0$ in $\{v>0\}\>,$ $\vert \nabla v \vert = 1$ on $\partial\{v>0\}.$ We show that blow-up limits of u and v are backward self-similar solutions and use this to determine the Hausdorff dimension of the free boundaries $\partial\{u>0\}$ and $\partial\{v>0\}.$

Exact solutions of the Navier-Stokes equations via Leray’s scheme

J. Leray considered a backward self-similar solution of the Navier-Stokes equations in the hope that it gives us an example of the finite-time blow-up of the three dimensional nonstationary Navier-Stokes equations. However, he showed no example of solutions. We list here some particular solutions and discuss their fluid mechanical properties. We also consider a forward self-similar solution, which describes solutions decaying as time tends to infinity.