Local Blow-up Research Articles

Local existence and blowup criterion for the Euler equations in a function space with nondecaying derivatives

We consider the Cauchy problem for the incompressible Euler equations on Rd for d≥3. Then we demonstrate the local-in-time solvability of classical solutions with the nondecaying derivatives and finite kinetic energy. Moreover, we establish the blowup criterion of such solutions in terms of the vorticity.

Stochastic MHD equations with fractional kinematic dissipation and partial magnetic diffusion in [formula omitted

This paper focuses on a system of the 2-D MHD equations with fractional kinematic dissipation and partial magnetic diffusion under random perturbations. We first establish the local existence, uniqueness and blow-up criterion of pathwise classical solution to the corresponding SPDE with general multiplicative noise in R2. Moreover, when the noise is non-autonomous and linear, we further establish global existence and large-time behavior of pathwise solutions by exploiting the structure of the equations and some estimates on Girsanov type stochastic processes.

Local well-posedness and blow-up for a family of U(1)-invariant peakon equations

The Cauchy problem for a unified family of integrable U(1)-invariant peakon equations from the NLS hierarchy is studied. As main results, local well-posedness is proved in Besov spaces, and blow-up is established through use of an L1 conservation law.

Local existence and blow-up criterion for the two and three dimensional ideal magnetic Benard problem

In this article, we consider the ideal magnetic Benard problem in both two and three dimensions and prove the existence and uniqueness of strong local-in-time solutions, in Hs for s > (n/2)+1, n = 2,3. In addition, a necessary condition is derived for singularity development with respect to the BMO-norm of the vorticity and electrical current, generalizing the Beale-Kato-Majda condition for ideal hydrodynamics. For more information see https://ejde.math.txstate.edu/Volumes/2020/91/abstr.html

Local existence and Serrin-type blow-up criterion for strong solutions to the radiation hydrodynamic equations

We consider the Cauchy problem for the three-dimensional, compressible radiation hydrodynamic equations. We establish the existence and uniqueness of local strong solutions for large initial data satisfying some compatibility condition. The initial density need not be positive and may vanish in an open set. Moreover, we establish a Serrin-type blow-up criterion, which is stated in terms of the velocity and density variables [Formula: see text] and is independent of the temperature and the radiation intensity.

Analytical-Numerical Study of Finite-Time Blow-up of the Solution to the Initial-Boundary Value Problem for the Nonlinear Klein–Gordon Equation

An analytical-numerical approach is used to study the finite-time blow-up of the solution to the initial boundary-value problem for the nonlinear Klein–Gordon equation. An analytical analysis yields an upper estimate for the blow-up time of the solution with an arbitrary positive initial energy. With the use of this a priori information, the blow-up process is numerically analyzed in more detail. It is shown that the numerical analysis of the blow-up of the solution makes it possible to improve the analytical estimate and to detect local blow-up with respect to the spatial variable.

Global Existence and Blow-Up for the Fractional p-Laplacian with Logarithmic Nonlinearity

In this paper, we study the following Dirichlet problem for a parabolic equation involving fractional p-Laplacian with logarithmic nonlinearity $$\begin{aligned} \left\{ \begin{array}{llc} u_{t}+(-\Delta )^{s}_{p}u+|u|^{p-2}u=|u|^{p-2}u\log (|u|) &{} \text {in}\ &{} \Omega ,\;t>0 , \\ u =0 &{} \text {in} &{} {\mathbb {R}}^{N}\backslash \Omega ,\;t > 0, \\ u(x,0)=u_{0}(x), &{} \text {in} &{}\Omega , \end{array}\right. \end{aligned}$$ where $$\Omega \subset {\mathbb {R}}^N \, ( N\ge 1)$$ is a bounded domain with Lipschitz boundary and $$2\le p< \infty $$ . The local existence will be done using the Galerkin approximations. By combining the potential well theory with the Nehari manifold, we establish the existence of global solutions. Then by virtue of a differential inequality technique, we prove that the local solutions blow-up in finite time with arbitrary negative initial energy and suitable initial values. Moreover, we give decay estimates of global solutions. The main difficulty here is the lack of logarithmic Sobolev inequality concerning fractional p-Laplacian.

On the initial and terminal value problem for a class of semilinear strongly material damped plate equations

We study the semilinear strongly damped plate equation by considering its two different problems. For initial value problem, we prove the local well-posedness and blow-up results of solution for the problem with polynomial nonlinear source terms. For terminal value problem, given the ill-posedness in the sense of Hadamard we propose a regularization method for the problem with logarithmic nonlinearities. Under the a priori assumption on the exact solution belonging to a Gevrey space, we propose the Fourier truncation method to stabilize the ill-posed problem, also by establishing some stability estimates of logarithmic type in Lq space.

On the diffusion p(x)-Laplacian with logarithmic nonlinearity

In this paper, we consider the following class of heat equation involving p(x)-Laplacian with logarithmic nonlinearity $$\begin{aligned}\left\{ \begin{array}{llc} u_{t}-\Delta _{p(x)}u=|u|^{s(x)-2}u\log (|u|) &{} \text {in}\ &{} \Omega ,\;t>0 , \\ u =0 &{} \text {in} &{} \partial \Omega ,\;t > 0, \\ u(x,0)=u_{0}(x), &{} \text {in} &{}\Omega , \end{array}\right. \end{aligned}$$ where $$\Omega \subset \mathbb {R}^{N}$$ ( $$N\ge 1$$ ) is a bounded domain with smooth boundary $$\partial \Omega $$ , $$p, s: \overline{\Omega }\rightarrow \mathbb {R}_{+}$$ are continuous functions that satisfy some technical conditions and $$-\Delta _{p(x)}$$ is the $$p(x)-$$ Laplacian, which generalizes the $$p-$$ Laplacian operator $$-\Delta _{p}$$ . The local existence will be done by using the Galerkin method. Then, by using the concavity method we prove that the local solutions blow-up in finite time under suitable conditions. In order to prove the global existence, we will use the potential well theory combined with the Pohozaev manifold that is a novelty for this type of problem. The difficulty here is the lack of logarithmic Sobolev inequality which seems there is no logarithmic Sobolev inequality concerning the p(x)-Laplacian yet.

Non-degeneracy of the ground state solution on nonlinear Schrödinger equation

We are concerned with the following nonlinear Schrödinger equation −ε2Δu+V(x)u=|u|p−2u,u∈H1(RN),where ε>0 is a small parameter, N≥1, 2<p<2∗. We prove the non-degeneracy of the ground state solution to the above problem by using local Pohozaev identity and blow-up analysis, which fills the gap in this aspect.

Local well-posedness and blow-up criteria for a three-component Camassa–Holm type equation

Considered herein is a three-component Camassa–Holm type equation that admits a bi-Hamiltonian structure and infinitely many conserved quantities. In this paper, we establish the local well-posedness of this system in Bp,rs. We also deduce two blow-up criteria for this system.

Local existence, global existence and blow-up of solutions to a nonlocal Kirchhoff diffusion problem**This work is supported by NSFC (Grant No. 11201380).

In this paper, we study the following diffusion model of Kirchhoff-type driven by a nonlocal integro-differential operator where [u]s is the Gagliardo seminorm of u, is a bounded domain with Lipschitz boundary, , is a nonlocal integro-differential operator defined in (), which generalizes the fractional Laplace operator , is the initial function, and is a continuous function and there exist two constants m0 > 0 and such that As is well-known, the nonlocal Kirchhoff problem was first introduced and motivated in Fiscella and Valdinoci (2014 Nonlinear Anal. 94 156–70) and the above problem was studied by Xiang et al (2018 Nonlinearity 31 3228–50), the main results of Xiang et al (2018 Nonlinearity 31 3228–50) are as follows: The local existence of nontrivial, nonnegative weak solution for , where . The blow-up conditions for nontrivial, nonnegative weak solution when J(u0) < 0, where J(u0) denotes the initial energy.The main purpose of this paper is to extend the above results and we get: The global existence of nontrivial, nonnegative weak solution for any . The global existence and blow-up conditions for nontrivial, nonnegative weak solution when for the case , where d is a positive constant given in ().

Local existence and blow-up of solutions for coupled viscoelastic wave equations with degenerate damping terms

Local existence and blow-up of solutions for coupled viscoelastic wave equations with degenerate damping terms

Navier–Stokes equations: local existence, uniqueness and blow-up of solutions in Sobolev–Gevrey spaces

ABSTRACT This work establishes local existence and uniqueness as well as blow-up criteria for solutions of the Navier–Stokes equations in Sobolev–Gevrey spaces . More precisely, if it is assumed that the initial data belongs to , with , we prove that there is a time T>0 such that for and . If the maximal time interval of existence of solutions is finite, , then, we prove, for example, that the blow-up inequality holds for , a>0, ( is the integer part of ).

Solutions with Prescribed Local Blow-up Surface for the Nonlinear Wave Equation

Abstract We prove that any sufficiently differentiable space-like hypersurface of ℝ 1 + N {{\mathbb{R}}^{1+N}} coincides locally around any of its points with the blow-up surface of a finite-energy solution of the focusing nonlinear wave equation ∂ t ⁢ t ⁡ u - Δ ⁢ u = | u | p - 1 ⁢ u {\partial_{tt}u-\Delta u=|u|^{p-1}u} on ℝ × ℝ N {{\mathbb{R}}\times{\mathbb{R}}^{N}} , for any 1 ≤ N ≤ 4 {1\leq N\leq 4} and 1 &lt; p ≤ N + 2 N - 2 {1&lt;p\leq\frac{N+2}{N-2}} . We follow the strategy developed in our previous work (2018) on the construction of solutions of the nonlinear wave equation blowing up at any prescribed compact set. Here to prove blow-up on a local space-like hypersurface, we first apply a change of variable to reduce the problem to blowup on a small ball at t = 0 {t=0} for a transformed equation. The construction of an appropriate approximate solution is then combined with an energy method for the existence of a solution of the transformed problem that blows up at t = 0 {t=0} . To obtain a finite-energy solution of the original problem from trace arguments, we need to work with H 2 × H 1 {H^{2}\times H^{1}} solutions for the transformed problem.

Removing Type II Singularities Off the Axis for the Three Dimensional Axisymmetric Euler Equations

In this paper we obtain new local blow-up criterion for smooth axisymmetric solutions to the three dimensional incompressible Euler equation. If the vorticity satisfies $$ \int \nolimits _{0}^{t_*} (t_*-t) \Vert \omega (t)\Vert _{ L^\infty (B(x_{ *}, R_0))}\,{\hbox {d}}t <+\infty $$ for a ball $$B(x_{ *}, R_0)$$ away from the axis of symmetry, then there exists no singularity at $$t=t_*$$ in the torus $$T(x_*, R)$$ generated by rotation of the ball $$B(x_{ *}, R_0)$$ around the axis. This implies that possible singularity at $$t=t_*$$ in the torus $$T(x_*, R)$$ is excluded if the vorticity satisfies the blow-up rate $$ \Vert \omega (t)\Vert _{L^\infty (T(x_*, R))}= O\left( \frac{1}{(t_*-t)^\gamma }\right) $$ as $$t\rightarrow t_*$$ , where $$\gamma <2$$ , and the torus $$T(x_*, R)$$ does not touch the axis.

On local existence, uniqueness and blow-up of solutions for the generalized MHD equations in Lei–Lin spaces

This paper establishes the existence and uniqueness, and also presents a specific blow-up criterion, for solutions of the generalized magnetohydrodynamics (GMHD) equations in Lei–Lin spaces $$\mathcal {X}^s(\mathbb {R}^3)$$ , by considering appropriate values for s. More precisely, if it is assumed that the initial data $$(u_0,b_0)$$ belong to $$\mathcal {X}^{s}(\mathbb {R}^3)$$ , we demonstrate that there exists an instant of time $$T>0$$ such that $$(u,b)\in [C_{T}(\mathcal {X}^s(\mathbb {R}^3))\cap L^1_{T}({\mathcal {X}}^{s+2\alpha }(\mathbb {R}^3))]\times [C_{T}(\mathcal {X}^s(\mathbb {R}^3))\cap L^1_{T}({\mathcal {X}}^{s+2\beta }(\mathbb {R}^3))]$$ , provided that $$\alpha ,\beta \in (\frac{1}{2},1]$$ and $$\max \big \{\frac{\alpha (1-2\beta )}{\beta },\frac{\beta (1-2\alpha )}{\alpha }\big \} \le s<0$$ (here $$\alpha $$ and $$\beta $$ are related to the fractional Laplacian that appears in the GMHD system). Furthermore, we prove that if $$T^*$$ (finite) is the first blow-up instant of the solution (u, b)(x, t), then $$ \lim _{t\nearrow T^*}\Vert (u,b)(t)\Vert _{\mathcal {X}^s(\mathbb {R}^3)}=\infty $$ , whether $$\max \big \{1-2\alpha ,1-2\beta ,\frac{\alpha (1-2\beta )}{\beta },\frac{\beta (1-2\alpha )}{\alpha }\big \}< s<0$$ and $$\alpha ,\beta \in (\frac{1}{2},1]$$ .

The local existence and blowup criterion for strong solutions to the kinetic Cucker–Smale model coupled with the compressible Navier–Stokes equations

In this paper, we establish the existence and uniqueness of local strong solutions to the kinetic Cucker–Smale model coupled with the isentropic compressible Navier–Stokes equations in the whole space. Moreover, the blowup mechanism for strong solutions to the coupled system is also investigated.

Local well-posedness and blow-up phenomena of the generalized short pulse equation

In this paper, we investigate the Cauchy problem of the generalized short-pulse equation in periodic domain. We first establish the local well-posedness of the equation in Hs(S),s≥2 by taking advantage of the Kato method and present some useful conservation laws. Then, we obtain the boundedness of the solution by virtue of the conservation laws. Finally, we give a precise blow-up criterion for α &amp;gt; 0 (or α &amp;lt; 0) and get several blow-up results provided the initial data satisfies some certain conditions.

Blow-up of radially symmetric solutions for a semilinear heat equation on hyperbolic space

In this paper radially symmetric solutions of a semilinear heat equation $$u_{t}=\Delta u + u^{p}$$ on the hyperbolic space are considered. First universal bounds of the nonnegative solution are obtained to know the blow-up rate at the final blow-up time under the exponent p which is subcritical in the Sobolev sense. Next we derive its local blow-up profile and also analyze blow-up set of solutions.