Backward Self-similar Solutions Research Articles

Singularity formation for Burgers' equation with transverse viscosity

We consider Burgers equation with transverse viscosity $$\partial_tu+u\partial_xu-\partial_{yy}u=0, (x,y)\in \mathbb R^2, u:[0,T)\times \mathbb R^2\rightarrow \mathbb R.$$ We construct and describe precisely a family of solutions which become singular in finite time by having their gradient becoming unbounded. To leading order, the solution is given by a backward self-similar solution of Burgers equation along the $x$ variable, whose scaling parameters evolve according to parabolic equations along the $y$ variable, one of them being the quadratic semi-linear heat equation. We develop a new framework adapted to this mixed hyperbolic/parabolic blow-up problem, revisit the construction of flat blow-up profiles for the semi-linear heat equation, and the self-similarity in the shocks of Burgers equation.

Leray's Backward Self-Similar Solutions to the 3D Navier--Stokes Equations in Morrey Spaces

In this paper, it is shown that there does not exist a non-trivial Leray's backward self-similar solution to the 3D Navier-Stokes equations with profiles in Morrey spaces $\dot{\mathcal{M}}^{q,1}(\mathbb{R}^{3})$ provided $3/2<q<6$, or in $\dot{\mathcal{M}}^{q,l}(\mathbb{R}^{3})$ provided $6\leq q<\infty$ and $2<l\leq q$. This generalizes the corresponding results obtained by Ne\v{c}as-R\r{a}u\v{z}i\v{c}ka-\v{S}ver\'{a}k [19, Acta.Math. 176 (1996)] in $L^{3}(\mathbb{R}^{3})$, Tsai [25, Arch. Ration. Mech. Anal. 143 (1998)] in $L^{p}(\mathbb{R}^{3})$ with $p\geq3$,, Chae-Wolf [3, Arch. Ration. Mech. Anal. 225 (2017)] in Lorentz spaces $L^{p,\infty}(\mathbb{R}^{3})$ with $p>3/2$, and Guevara-Phuc [11, SIAM J. Math. Anal. 12 (2018)] in $\dot{\mathcal{M}}^{q,\frac{12-2q}{3}}(\mathbb{R}^{3})$ with $12/5\leq q<3$ and in $L^{q, \infty}(\mathbb{R}^3)$ with $12/5\leq q<6$.

Backward self-similar solutions for compressible Navier–Stokes equations

This article is devoted to backward self-similar blow up solutions of the compressible Navier–Stokes equations with radial symmetry. We show that such solutions cannot exist if they either have finite energy, or satisfy appropriate smallness conditions. Furthermore, numerical simulations lead us to the conjecture that such solutions do not exist.

Self-similar shrinkers of the one-dimensional Landau–Lifshitz–Gilbert equation

The main purpose of this paper is the analytical study of self-shrinker solutions of the one-dimensional Landau–Lifshitz–Gilbert equation (LLG), a model describing the dynamics for the spin in ferromagnetic materials. We show that there is a unique smooth family of backward self-similar solutions to the LLG equation, up to symmetries, and we establish their asymptotics. Moreover, we obtain that in the presence of damping, the trajectories of the self-similar profiles converge to great circles on the sphere $$\mathbb {S}^2$$ , at an exponential rate. In particular, the results presented in this paper provide examples of blow-up in finite time, where the singularity develops due to rapid oscillations forming limit circles.

On the multiplicity of self-similar solutions of the semilinear heat equation

In studies of superlinear parabolic equations ut=Δu+up,x∈RN,t>0,where p>1, backward self-similar solutions play an important role. These are solutions of the form u(x,t)=(T−t)−1∕(p−1)w(y), where y≔x∕T−t, T is a constant, and w is a solution of the equation Δw−y⋅∇w∕2−w∕(p−1)+wp=0. We consider (classical) positive radial solutions w of this equation. Denoting by pS, pJL, pL the Sobolev, Joseph-Lundgren, and Lepin exponents, respectively, we show that for p∈(pS,pJL) there are only countably many solutions, and for p∈(pJL,pL) there are only finitely many solutions. This result answers two basic open questions regarding the multiplicity of the solutions.

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.

Leray's Self-Similar Solutions to the Navier--Stokes Equations with Profiles in Marcinkiewicz and Morrey Spaces

We rule out the existence of Leray's backward self-similar solutions to the Navier--Stokes equations with profiles in $L^{12/5}({R}^3)$ or in the Marcinkiewicz space $L^{q,\infty}({R}^3)$ for $q\in(12/5, 6)$. This follows from a more general result formulated in terms of Morrey spaces and the first-order Riesz potential.

Blow-up behavior of solutions to a degenerate parabolic–parabolic Keller–Segel system

Qualitative properties of solutions blowing up in finite time are obtained for a degenerate parabolic–parabolic Keller–Segel system, the nonlinear diffusion being of porous medium type with an exponent smaller or equal to the critical one $$m_c:=2(N-1)/N$$ . In both cases, it is shown that only type II blow-up is possible, that is, blow-up at the same rate as backward self-similar solutions never occurs. Further information on the generation of singularities induced by mass concentration are given in the critical case.

On the dead-core problem for the $p$-Laplace equation with a strong absorption

We study an initial boundary value problem for the $p$-Laplace equation with a strong absorption. We are concerned with the dead-core behavior of the solution. First, some criteria for developing dead-core are given. Also, the temporal dead-core rate for certain initial data is determined. Then we prove uniqueness theorem for the backward self-similar solutions.

On the locally self-similar singular solutions for the incompressible Euler equations

In this paper we consider the locally backward self-similar solutions for the Euler system, and focus on the case that the possible nontrivial velocity profiles have non-decaying asymptotics. We derive the meaningful representation formula of the pressure profile in terms of velocity profiles in this case, and by using it and the local energy inequality of profiles, we prove some nonexistence results and show the energy behavior concerning the possible velocity profiles.

Non-accessible singular homoclinic orbits for a semilinear parabolic equation

We show the existence of at least three different continuations beyond blow-up for a backward self-similar solution of a supercritical Fujita equation. One of these extended solutions cannot be approximated by classical entire solutions in a specific way given by the scaling invariance of the equation, while the minimal continuation is known to be accessible by a family of such entire solutions.

Asymptotic behavior at infinity of the stationary solution to a semilinear heat equation

In this paper we investigate the asymptotic behavior at infinity of the backward self-similar solution of the differential equation ut=Δu+eu, x∈Ω,t>0, where Ω is a ball with the Dirichlet boundary or Rn, 3≤n<∞. We prove that, under some reasonable condition at infinity, every radial symmetric, nontrivial, bounded above solution of the equationωyy+(n−1y−y2)ωy+eω−1=0 tends to minus infinity as y→∞. This equation comes from the scaled ignition model. Furthermore, ω+logy2 converges to a constant for sufficiently large y. This result extends the similar one in Lacey (1993) for an arbitrary solution which is bounded above and for dimension 3≤n<∞ in space.

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.

Blow-up rate for radially symmetric solutions of some parabolic equations with nonlinear boundary conditions

We consider the following parabolic equations with nonlinear boundary conditions: ut=Δu−a|u|p−1u in B1×(0,T), ∂νu=|u|q−1u on ∂B1×(0,T), where p,q>1, a⩾0 and B1 is the unit ball. We study the blow-up rate of radial symmetric solutions u(r,t) without any assumptions on the initial data. Furthermore we show the uniqueness of positive solutions of one dimensional backward self-similar solutions. As a consequence, we can determine the asymptotic behavior of blow-up solutions near the blow-up time.

Non-existence of radial backward self-similar blow-up solutions with sign change

We consider a Cauchy problem for a semilinear heat equationwith p &gt; 1. If u(x, t) = (T − t)−1/(p−1)ϕ((T − t)−1/2x) for x ∈ ℝN and t ∈ [0, T),where ϕ ∈ L∞(ℝN) is a solution not identically equal to zero ofthen u is called a backward self-similar solution blowing up at t = T. We show that, for all p &gt; 1, there exists no radial sign-changing solution of (E) which belongs to L∞(ℝN). This implies the non-existence of radial backward self-similar solution with sign change blowing up in finite time.

Singular backward self-similar solutions of a semilinear parabolic equation

We consider a parabolic partial differential equation with power nonlinearity. Our concern is the existence of a singular solution whose singularity becomes anomalous in finite time. First we study the structure of singular radial solutions for an equation derived by backward self-similar variables. Using this, we obtain a singular backward self-similar solution whose singularity becomes stronger or weaker than that of a singular steady state.

On backward self-similar blow-up solutions to a supercritical semilinear heat equation

We are concerned with a Cauchy problem for the semilinear heat equationthen u is called a backward self-similar solution blowing up at t = T. Let pS and pL be the Sobolev and the Lepin exponents, respectively. It was shown by Mizoguchi (J. Funct. Analysis257 (2009), 2911–2937) that k ≡ (p − 1)−1/(p−1) is a unique regular radial solution of (P) if p &gt; pL. We prove that it remains valid for p = pL. We also show the uniqueness of singular radial solution of (P) for p &gt; pS. These imply that the structure of radial backward self-similar blow-up solutions is quite simple for p ≥ pL.

Nonexistence of backward self-similar blowup solutions to a supercritical semilinear heat equation

We consider a Cauchy problem for a semilinear heat equation { u t = Δ u + u p in R N × ( 0 , T ) , u ( x , 0 ) = u 0 ( x ) ⩾ 0 in R N with p > p S where p S is the Sobolev exponent. If u ( x , t ) = ( T − t ) − 1 / ( p − 1 ) φ ( ( T − t ) − 1 / 2 x ) for x ∈ R N and t ∈ [ 0 , T ) , where φ is a regular positive solution of (P) Δ φ − y 2 ∇ φ − 1 p − 1 φ + φ p = 0 in R N , then u is called a backward self-similar blowup solution. It is immediate that (P) has a trivial positive solution κ ≡ ( p − 1 ) − 1 / ( p − 1 ) for all p > 1 . Let p L be the Lepin exponent. Lepin obtained a radial regular positive solution of (P) except κ for p S < p < p L . We show that there exist no radial regular positive solutions of (P) which are spatially inhomogeneous for p > p L .

On the self-similar solutions of the magneto-hydro-dynamic equations

In this paper, we show that, for the three dimensional incompressible magneto-hydro-dynamic equations, there exists only trivial backward self-similar solution in Lp(M.3) for p ≥ 3, under some smallness assumption on either the kinetic energy of the self-similar solution related to the velocity field, or the magnetic field. Second, we construct a class of global unique forward self-similar solutions to the three-dimensional MHD equations with small initial data in some sense, being homogeneous of degree −1 and belonging to some Besov space, or the Lorentz space or pseudo-measure space, as motivated by the work in [5].

Backward selfsimilar solutions of supercritical parabolic equations

We consider the exponential reaction–diffusion equation in space-dimension n ∈ ( 2 , 10 ) . We show that for any integer k ≥ 2 there is a backward selfsimilar solution which crosses the singular steady state k -times. The same holds for the power nonlinearity if the exponent is supercritical in the Sobolev sense and subcritical in the Joseph–Lundgren sense.