Forward Self-similar Solutions Research Articles

Self-similar shrinking of supports and non-extinction for a nonlinear diffusion equation with spatially inhomogeneous strong absorption

We study the dynamics of the following porous medium equation with strong absorption [Formula: see text] posed for [Formula: see text], with [Formula: see text], [Formula: see text] and [Formula: see text]. Considering the Cauchy problem with non-negative initial condition [Formula: see text], instantaneous shrinking and localization of supports for the solution [Formula: see text] at any [Formula: see text] are established. With the help of this property, existence and uniqueness of a non-negative compactly supported and radially symmetric forward self-similar solution with algebraic decay in time are proven. Finally, it is shown that finite time extinction does not occur for a wide class of initial conditions and this unique self-similar solution is the pattern for large time behavior of these general solutions.

Optimal well-posedness and forward self-similar solution for the Hardy–Hénon parabolic equation in critical weighted Lebesgue spaces

The Cauchy problem for the Hardy–Hénon parabolic equation is studied in the critical and subcritical weighted Lebesgue spaces on the Euclidean space Rd. In earlier works, the well-posedness of singular initial data and the existence of non-radial forward self-similar solutions to the problem were shown for the Hardy and Fujita cases (γ≤0). The weighted spaces are used to treat the potential |x|γ as an increase or decrease in the weight, which enables us to prove the well-posedness of the problem for all γ, with −min{2,d}<γ, including the Hénon case (γ>0). As a by-product of global existence, self-similar solutions to the problem are established for all γ without restrictions. Furthermore, the non-existence of a local solution for supercritical data is also shown. Therefore, our critical exponent, sc is optimal with regard to solvability.

Self-similar source-type solutions to the three-dimensional Navier–Stokes equations

We formalize a systematic method of constructing forward self-similar solutions to the Navier–Stokes equations in order to characterize the late stage of decaying process of turbulent flows. (i) In view of critical scale-invariance of type 2 we exploit the vorticity curl as the dependent variable to derive and analyse the dynamically scaled Navier–Stokes equations. This formalism offers the viewpoint from which the problem takes the simplest possible form. (ii) Rewriting the scaled Navier–Stokes equations by Duhamel principle as integral equations, we regard the nonlinear term as a perturbation using the Fokker–Planck evolution semigroup. Systematic successive approximations are introduced and the leading-order solution is worked out explicitly as the Gaussian function with a solenoidal projection. (iii) By iterations the second-order approximation is estimated explicitly up to solenoidal projection and is evaluated numerically. (iv) A new characterization of nonlinear term is introduced on this basis to estimate its strengthNquantitatively. We find thatN=O(10−2)for the three-dimensional Navier–Stokes equations. This should be contrasted withN=O(10−1)for the Burgers equations andN≡0for the two-dimensional Navier–Stokes equations. (v) As an illustration we explicitly determine source-type solutions to the multi-dimensional Burgers equations. Implications and applications of the current results are given.

A Liouville theorem for ancient solutions to a semilinear heat equation and its elliptic counterpart

&lt;p style='text-indent:20px;'&gt;We establish the nonexistence of nontrivial ancient solutions to the nonlinear heat equation &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$ u_t = \Delta u+|u|^{p-1}u $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; which are smaller in absolute value than the self-similar radial singular steady state, provided that the exponent &lt;inline-formula&gt;&lt;tex-math id="M2"&gt;\begin{document}$ p $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; is strictly between Serrin's exponent and that of Joseph and Lundgren. This result was previously established by Fila and Yanagida [Tohoku Math. J. (2011)] by using forward self-similar solutions as barriers. In contrast, we apply a sweeping argument with a family of time independent weak supersolutions. Our approach naturally lends itself to yield an analogous Liouville type result for the steady state problem in higher dimensions. In fact, in the case of the critical Sobolev exponent we show the validity of our results for solutions that are smaller in absolute value than a 'Delaunay'-type singular solution.&lt;/p&gt;

An approximation of forward self-similar solutions to the 3D Navier-Stokes system

In this paper, we present two constructions of forward self-similar solutions to the $3$D incompressible Navier-Stokes system, as the singular limit of forward self-similar solutions to certain parabolic systems.

New critical exponents for a doubly singular parabolic equation

ABSTRACT This paper deals with the Cauchy problem for a doubly singular parabolic equation with nonlocal inner source where , , , q>1, , and r + s>1. We first obtain a new critical Fujita exponent by virtue of the auxiliary function method and the forward self-similar solution, and then determine the second critical exponent to classify global and non-global solutions of the problem in the coexistence region via the decay rates of an initial data at spatial infinity. Moreover, the large time behavior of global solution and the life span of non-global solution are derived.

Finite-time blowup in Cauchy problem of parabolic-parabolic chemotaxis system

This paper is concerned with blowup in a parabolic-parabolic system describing chemotactic aggregation. In a disk, radial solutions blow up in finite time if their initial energy is less than some value. In the whole plane, the energy diverges to −∞ as time goes to +∞ for any forward selfsimilar solution. This implies that one cannot expect to get a sufficient condition for finite-time blowup using energy as in a disk. For a solution (u,v), u and v denote density of cells and of chemical substance, respectively. Let τ be the coefficient of time derivative of v. We first prove that for τ>0 there exists M(τ)>0 with M(τ)→∞ as τ→∞ such that all radial solutions (u,v) with initial mass of u larger than M(τ) blow up in finite time. On the other hand, it was shown in [22] that any blowup in the system with τ=1 is type II (not necessarily in radial case). Removing the restriction on τ, we get the conclusion for all τ>0.

New critical exponents, large time behavior, and life span for a fast diffusive p-Laplacian equation with nonlocal source

This paper mainly investigate positive solutions of the Cauchy problem for a fast diffusive p-Laplacian equation with nonlocal source $$\begin{aligned} u_{t}=\Delta _pu+\left( \,\,\int \limits _{{\mathbb {R}}^N}u^q(y,t)\mathrm{d}y\right) ^{\frac{r-1}{q}}u^{s+1},\quad (x,t)\in {\mathbb {R}}^N\times (0,T), \end{aligned}$$ where $$N\ge 1$$ , $$\frac{2N}{N+1}<p<2$$ , $$q>1$$ , $$r\ge 1$$ , $$0\le s<\left( 1+\frac{1}{N}\right) p-2$$ and $$r+s>1$$ . We obtain the new critical Fujita exponent by virtue of the auxiliary function method and forward self-similar solution, and then determine the second critical exponent to classify global and non-global solutions of the problem in the coexistence region via the decay rates of an initial data at spatial infinity. Moreover, the large time behavior of global solution and the life span of non-global solution are derived.

Forward self-similar solutions of the fractional Navier-Stokes equations

We study forward self-similar solutions to the 3-D Navier-Stokes equations with the fractional diffusion (−Δ)α. First, we construct a global-time forward self-similar solutions to the fractional Navier-Stokes equations with 5/6<α≤1 for arbitrarily large self-similar initial data by making use of the so called blow-up argument. Moreover, we prove that this solution is smooth in R3×(0,+∞). In particular, when α=1, we prove that the solution constructed by Korobkov and Tsai (2016) [16] satisfies the decay estimate by establishing regularity of solution for the corresponding elliptic system, which implies this solution has the same properties as a solution which was constructed in Jia and Šverák (2014) [13].

Asymptotically self-similar behaviour of global solutions for semilinear heat equations with algebraically decaying initial data

AbstractWe consider the Cauchy problem $$\left\{ {\matrix{ {u_t = \Delta u + u^p,\quad } \hfill &amp; {x\in {\bf R}^N,\;t \leq 0,} \hfill \cr {u(x,0) = u_0(x),\quad } \hfill &amp; {x\in {\bf R}^N,} \hfill \cr } } \right.$$where N &gt; 2, p &gt; 1, and u0 is a bounded continuous non-negative function in RN. We study the case where u0(x) decays at the rate |x|−2/(p−1) as |x| → ∞, and investigate the convergence property of the global solutions to the forward self-similar solutions. We first give the precise description of the relationship between the spatial decay of initial data and the large time behaviour of solutions, and then we show the existence of solutions with a time decay rate slower than the one of self-similar solutions. We also show the existence of solutions that behave in a complicated manner.

Existence of peaking solutions for semilinear heat equations with blow-up profile above the singular steady state

We consider positive solutions of the semilinear heat equation with supercritical power nonlinearity, and construct peaking solutions by connecting a backward self-similar solution with a forward self-similar solution. In particular, we show the existence of incomplete blow-up solutions with blow-up profile above the singular steady state. We also show the existence of a minimal continuation and obtain some properties of non-minimal continuations.

Global existence and blow-up of solutions for the heat equation with exponential nonlinearity

We are concerned with the existence of global in time solution for a semilinear heat equation with exponential nonlinearity(P){∂tu=Δu+eu,x∈RN,t>0,u(x,0)=u0(x),x∈RN, where u0 is a continuous initial function. In this paper, we consider the case where u0 decays to −∞ at space infinity, and study the optimal decay bound classifying the existence of global in time solutions and blowing up solutions for (P). In particular, we point out that the optimal decay bound for u0 is related to the decay rate of forward self-similar solutions of ∂tu=Δu+eu.

Forward Self-Similar Solutions to the Viscoelastic Navier--Stokes Equation with Damping

Motivated by [H. Jia and V. Šverák, Invent. Math., 196 (2014), pp. 233--265], we prove that there exists a global, forward self-similar solution to the viscoelastic Navier--Stokes equation with damping, that is smooth for $t>0$, for any initial data that is homogeneous of degree -1.

Forward self-similar solutions of the Navier–Stokes equations in the half space

For the incompressible Navier-Stokes equations in the 3D half space, we show the existence of forward self-similar solutions for arbitrarily large self-similar initial data.

Forward Discretely Self-Similar Solutions of the Navier–Stokes Equations II

For any discretely self-similar, incompressible initial data which are arbitrarily large in weak $$L^3$$ , we construct a forward discretely self-similar solution to the 3D Navier–Stokes equations in the whole space. This also gives a third construction of self-similar solutions for any $$-1$$ -homogeneous initial data in weak $$L^3$$ , improving those in JiaSverak and Sverak (Invent Math 196(1):233–265, 2014) and Korobkov and Tsai (Forward self-similar solutions of the Navier–Stokes equations in the half space, arXiv:1409.2516 , 2016) for Holder continuous data. Our method is based on a new, explicit a priori bound for the Leray equations.

Local-in-space estimates near initial time for weak solutions of the Navier-Stokes equations and forward self-similar solutions

We show that the classical Cauchy problem for the incompressible 3d Navier-Stokes equations with (−1)-homogeneous initial data has a global scale-invariant solution which is smooth for positive times. Our main technical tools are local-in-space regularity estimates near the initial time, which are of independent interest.

Self-Similar Solutions of the Compressible Flow in One-Space Dimension

For the isentropic compressible fluids in one-space dimension, we prove that the Navier-Stokes equations with density-dependent viscosity have neither forward nor backward self-similar strong solutions with finite kinetic energy. Moreover, we obtain the same result for the nonisentropic compressible gas flow, that is, for the fluid dynamics of the Navier-Stokes equations coupled with a transport equation of entropy. These results generalize those in Guo and Jiang's work (2006) where the one-dimensional compressible fluids with constant viscosity are considered.

The role of forward self-similar solutions in the Cauchy problem for semilinear heat equations

We consider the Cauchy problem{ut=Δu+up,x∈RN,t>0,u(x,0)=u0(x),x∈RN, where N>2, p>1, and u0 is a bounded continuous non-negative function in RN. We study the case where u0(x) decays at the rate |x|−2/(p−1) as |x|→∞, and investigate the stability and instability properties of forward self-similar solutions. In particular, we obtain optimal conditions on the initial function u0 for the global existence in terms of self-similar solutions, and show the asymptotically self-similar behavior of the global solutions. We also obtain the condition for finite time blow-up by making use of the behavior of u0(x) as |x|→∞.

Blowup and self-similar solutions for two-component drift–diffusion systems

We discuss asymptotic properties of solutions of two-component parabolic drift–diffusion systems coupled through an elliptic equation in two space dimensions. In particular, conditions for finite time blowup versus the existence of forward self-similar solutions are studied.

A singular solution with smooth initial data for a semilinear parabolic equation

We consider the Cauchy problem for a parabolic partial differential equation with a power nonlinearity. Our concern in this paper is the existence of a singular solution with smooth initial data. By using the Haraux–Weissler equation, it is shown that there exist singular forward self-similar solutions. Using this result, we also obtain a sufficient condition for the singular solution with general initial data including smooth initial data.