Finite Time Blow-up Of Solutions Research Articles

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 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.

Self-Similar Blow-Up Profiles for a Reaction-Diffusion Equation with Strong Weighted Reaction

Abstract We study the self-similar blow-up profiles associated to the following second-order reaction-diffusion equation with strong weighted reaction and unbounded weight: ∂ t ⁡ u = ∂ x ⁢ x ⁡ ( u m ) + | x | σ ⁢ u p , \partial_{t}u=\partial_{xx}(u^{m})+|x|^{\sigma}u^{p}, posed for x ∈ ℝ {x\in\mathbb{R}} , t ≥ 0 {t\geq 0} , where m &gt; 1 {m&gt;1} , 0 &lt; p &lt; 1 {0&lt;p&lt;1} and σ &gt; 2 ⁢ ( 1 - p ) m - 1 {\sigma&gt;\frac{2(1-p)}{m-1}} . As a first outcome, we show that finite time blow-up solutions in self-similar form exist for m + p &gt; 2 {m+p&gt;2} and σ in the considered range, a fact that is completely new: in the already studied reaction-diffusion equation without weights there is no finite time blow-up when p &lt; 1 {p&lt;1} . We moreover prove that, if the condition m + p &gt; 2 {m+p&gt;2} is fulfilled, all the self-similar blow-up profiles are compactly supported and there exist two different interface behaviors for solutions of the equation, corresponding to two different interface equations. We classify the self-similar blow-up profiles having both types of interfaces and show that in some cases global blow-up occurs, and in some other cases finite time blow-up occurs only at space infinity. We also show that there is no self-similar solution if m + p &lt; 2 {m+p&lt;2} , while the critical range m + p = 2 {m+p=2} with σ &gt; 2 {\sigma&gt;2} is postponed to a different work due to significant technical differences.

On the Lagrangian Trajectories for the One-Dimensional Euler Alignment Model without Vacuum Velocity

A well-known result of Carrillo, Choi, Tadmor, and Tan [1] states that the 1D Euler Alignment model with smooth interaction kernels possesses a “critical threshold” criterion for the global existence or finite-time blowup of solutions, depending on the global nonnegativity (or lack thereof) of the quantity e 0 =∂ x u 0 +ϕ*ρ 0 . In this note, we rewrite the 1D Euler Alignment model as a first-order system for the particle trajectories in terms of a certain primitive ψ 0 of e 0 ; using the resulting structure, we give a complete characterization of global-in-time existence versus finite-time blowup of regular solutions that does not require a velocity to be defined in the vacuum. We also prove certain upper and lower bounds on the separation of particle trajectories, valid for smooth and weakly singular kernels, and we use them to weaken the hypotheses of Tan [25] sufficient for the global-in-time existence of a solution in the weakly singular case, when the order of the singularity lies in the range s∈(0,1 2) and the initial data is critical.

Blow-up and global existence for solutions to the porous medium equation with reaction and slowly decaying density

We study existence of global solutions and finite time blow-up of solutions to the Cauchy problem for the porous medium equation with a variable density ρ(x) and a power-like reaction term ρ(x)up with p>1; this is a mathematical model of a thermal evolution of a heated plasma (see [29]). The density decays slowly at infinity, in the sense that ρ(x)≲|x|−q as |x|→+∞ with q∈[0,2). We show that for large enough initial data, solutions blow-up in finite time for any p>1. On the other hand, if the initial datum is small enough and p>p¯, for a suitable p¯ depending on ρ,m,N, then global solutions exist. In addition, if p<p_, for a suitable p_≤p¯ depending on ρ,m,N, then the solution blows-up in finite time for any nontrivial initial datum; we need the extra hypothesis that q∈[0,ϵ) for ϵ>0 small enough, when m≤p<p_. Observe that p_=p‾, if ρ(x) is a multiple of |x|−q for |x| large enough. Such results are in agreement with those established in [48], where ρ(x)≡1, and are related to some results in [32,33]. The case of fast decaying density at infinity, i.e. q≥2, is examined in [36].

Instantaneous and finite time blow-up of solutions to a reaction-diffusion equation with Hardy-type singular potential

We deal with radially symmetric solutions to the reaction-diffusion equation with Hardy-type singular potentialut=Δum+K|x|2um, posed in RN×(0,T), in dimension N≥3, where m>1 and 0<K<(N−2)2/4. We prove that, in dependence of the initial condition u0∈L∞(RN)∩C(RN), its solutions may either blow up instantaneously or blow up in finite time at the origin, thus developing a singularity at x=0, but they can be continued globally in weak sense. The instantaneous blow-up occurs for example for any data u0 such that u0(0)>0. The proofs are based on a transformation mapping solutions to our equation into solutions to a non-homogeneous porous medium equation.

Blow-up of the radially symmetric solutions for the quadratic nonlinear Schrödinger system without mass-resonance

We consider the quadratic nonlinear Schrödinger system i∂tu+Δu=vu¯,i∂tv+κΔv=u2,onI×Rd,where 1≤d≤6 and κ>0. In the lower dimensional case d=1,2,3, it is known that the H1-solution is global in time. On the other hand, there are finite time blow-up solutions when d=4,5,6 and κ=1∕2. The condition of κ=1∕2 is called mass-resonance. In this paper, we prove finite time blow-up under radially symmetric assumption when d=5,6 and κ≠1∕2 and we show blow-up or grow-up when d=4.

Critical and sup-critical initial energy finite time blowup phenomena for the fourth-order wave equations with nonlinear strain term

This paper is devoted to the blowup phenomena for a class of fourth-order wave equation with nonlinear strain term. First, for the model equation with strain structure, we prove the finite time blowup of solution for the critical initial energy, i.e., E(0)=d. Then, the finite time blowup and the upper bound of the blowup time are established for the sup-critical initial energy, i.e., E(0)>0.

Blow-up profiles in quasilinear fully parabolic Keller–Segel systems

We examine finite-time blow-up solutions toin a ball , , where D and S generalize the functionswith . We show that if as well as and is a nonnegative, radially symmetric classical solution to () blowing up at , then there exists a so-called blow-up profile satisfyingMoreover, for all withwe can find such thatfor all .

Global existence and blowup solutions for the Gierer–Meinhardt system

We discuss global and blowup solutions for general Gierer–Meinhardt system with zero Neumann boundary conditions, which originally arose in studies of pattern-formation in biology and has interesting and challenging mathematical properties. We obtain some new sufficient conditions for global existence and finite time blow-up of solutions. In some special cases, the results are sharp.

Global well-posedness, asymptotic behavior and blow-up of solutions for a class of degenerate parabolic equations

The initial–boundary value problem for a class of degenerate parabolic equations is studied. Some new results on global existence of solutions are established by introducing a family of potential wells. In addition, asymptotic behavior and finite time blow-up of solutions are obtained in the case of subcritical initial energy and critical initial energy, respectively.

Optimal critical mass for the two-dimensional Keller–Segel model with rotational flux terms

Our aim is to show that several important systems of partial differential equations arising in mathematical biology, fluid dynamics and electrokinetics can be approached within a single model, namely, a Keller-Segel-type system with rotational flux terms. In particular, we establish sharp conditions on the optimal critical mass for having global existence and finite time blow-up of solutions in two spatial dimensions. Our results imply that the rotated chemotactic response can delay or even avoid the blow-up. The key observation is that for any angle of rotation α∈(-π, π], the resulting PDE system preserves a dissipative energy structure. Inspired by this property, we also provide an alternative derivation of the general system via an energetic variational approach. ©2020 International Press.

Finite time blow-up and global solutions for a nonlocal parabolic equation with Hartree type nonlinearity

This paper is concerned with the initial boundary value problem of a nonlocal parabolic equation. By establishing the comparison principle and studying the long-time behavior of its flow, we find the criteria for finite time blow-up and global existence of solutions respectively, which in particular includes the results of arbitrarily high energy initial data. We also characterize the asymptotic profile to both solutions vanishing at infinity and blowing up in finite time.

Finite time blow-up and global existence of solutions for semilinear parabolic equations with nonlinear dynamical boundary condition

&lt;p style='text-indent:20px;'&gt;For a class of semilinear parabolic equations under nonlinear dynamical boundary conditions in a bounded domain, we obtain finite time blow-up solutions when the initial data varies in the phase space &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;$ H_0^1(\Omega) $&lt;/tex-math&gt;&lt;/inline-formula&gt; at positive initial energy level and get global solutions with the initial data at low and critical energy level. Our main tools are potential well method and concavity method.&lt;/p&gt;

Initial-boundary value problem for a fourth-order plate equation with Hardy-Hénon potential and polynomial nonlinearity

&lt;p style='text-indent:20px;'&gt;In this paper, the initial-boundary value problem for a fourth-order plate equation with Hardy-Hénon potential and polynomial nonlinearity is invsitgated. First, we establish the local well-posedness of solutions by means of the semigroup theory. Then by using ordinary differential inequalities, potential well theory and energy estimate, we study the conditions on global existence and finite time blow-up. Moreover, the lifespan (i.e., the upper bound of the blow-up time) of the finite time blow-up solution is estimated.&lt;/p&gt;

Global existence and blow-up of solutions to a semilinear heat equation with logarithmic nonlinearity

ABSTRACT In this paper, we study the initial boundary value problem of a semilinear heat equation with logarithmic nonlinearity. By using potential well method and energy method, we obtain the existence of global solutions and finite-time blow-up solutions. Moreover, the upper bound of blow-up time is discussed under suitable assumptions.

Blow-up for the sixth-order multidimensional generalized Boussinesq equation with arbitrarily high initial energy

In this paper, we consider the Cauchy problem for the sixth-order multidimensional generalized Boussinesq equation with double damping terms. By using the improved convexity method combined with Fourier transform, we show the finite time blow-up of solution with arbitrarily high initial energy.

Finite time blowup of solutions to semilinear wave equation in an exterior domain

We consider the initial-boundary value problem of semilinear wave equation with nonlinearity |u|p in exterior domain in RN(N≥3). Especially, the lifespan of blowup solutions with small initial data are studied. The result gives upper bounds of lifespan which is essentially the same as the Cauchy problem in RN. At least in the case N=4, their estimates are sharp in view of the work by Zha–Zhou [21]. The idea of the proof is to use special solutions to linear wave equation with Dirichlet boundary condition which are constructed via an argument based on Wakasa–Yordanov [15].

Critical mass for a two-species chemotaxis model with two chemicals in **Supported by the Key Project of Education Department of Liaoning Province (Grant No. LZD201701), the Science Foundation of Liaoning Education Department (Grant No. LYB201601) and the Fundamental Research Funds for Central Universities (Grant No. 3132019178).

In this paper, we study a two-species chemotaxis model with two chemicals in . Let be the initial mass of the two species respectively. The critical mass of the model is established as a curve of the form . That is to say the solutuons exist globally if , and the finite time blow-up of solutions may occur if .

Global existence and blow-up of solutions to a semilinear heat equation with singular potential and logarithmic nonlinearity

This paper concerns with a semilinear heat equation with singular potential and logarithmic nonlinearity. By using the logarithmic Sobolev inequality and a family of potential wells, the existence of global solutions and infinite time blow-up solutions are obtained. The results of this paper indicate that the polynomial nonlinearity is a critical condition of existence of finite time blow-up solutions to semilinear heat equation with singular potential.