### A quasilinear parabolic system with nonlocal boundary condition.

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This paper is dedicated to a regularity criterion for the 2D MHD equations and viscoelastic equations. We prove that if the magnetic field B, respectively the local deformation gradient F, satisfies $\nabla B,\nabla F\in {L}^{q}(0,T;{L}^{p}\left(\mathbb{R}\xb2\right))$ for 1/p + 1/q = 1 and 2 < p ≤ ∞, then the corresponding local solution can be extended beyond time T.

These notes present the main results of [22, 23, 24] concerning the mass critical (gKdV) equation ${u}_{t}+{({u}_{xx}+{u}^{5})}_{x}=0$ for initial data in ${H}^{1}$ close to the soliton. These works revisit the blow up phenomenon close to the family of solitons in several directions: definition of the stable blow up and classification of all possible behaviors in a suitable functional setting, description of the minimal mass blow up in ${H}^{1}$, construction of various exotic blow up rates in ${H}^{1}$, including grow up in infinite time.

We consider the critical nonlinear Schrödinger equation $i{u}_{t}=-\Delta u-{\left|u\right|}^{\frac{4}{N}}u$ with initial condition $u(0,x)={u}_{0}$ in dimension $N$. For ${u}_{0}\in {H}^{1}$, local existence in time of solutions on an interval $[0,T)$ is known, and there exists finite time blow up solutions, that is ${u}_{0}$ such that ${lim}_{t\to T\<+\infty}{\left|{u}_{x}\left(t\right)\right|}_{{L}^{2}}=+\infty $. This is the smallest power in the nonlinearity for which blow up occurs, and is critical in this sense. The question we address is to understand the blow up dynamic. Even though there exists an explicit example of blow up solution and a class of initial data...

We consider the mass critical (gKdV) equation ${u}_{t}+{({u}_{xx}+{u}^{5})}_{x}=0$ for initial data in ${H}^{1}$. We first prove the existence and uniqueness in the energy space of a minimal mass blow up solution and give a sharp description of the corresponding blow up soliton-like bubble. We then show that this solution is the universal attractor of all solutions near the ground state which have a defocusing behavior. This allows us to sharpen the description of near soliton dynamics obtained in [29].

This paper is mainly concerned with the blow-up and global existence profile for the Cauchy problem of a class of fully nonlinear degenerate parabolic equations with reaction sources.

This paper deals with the blow-up properties of positive solutions to a localized singular parabolic equation with weighted nonlocal nonlinear boundary conditions. Under certain conditions, criteria of global existence and finite time blow-up are established. Furthermore, when q=1, the global blow-up behavior and the uniform blow-up profile of the blow-up solution are described; we find that the blow-up set is the whole domain [0,a], including the boundary, in contrast to the case of parabolic equations...

We give a sufficient condition under which the solutions of the energy-critical nonlinear wave equation and Schrödinger equation with inverse-square potential blow up. The method is a modified variational approach, in the spirit of the work by Ibrahim et al. [Anal. PDE 4 (2011), 405-460].

The focusing nonlinear Schrödinger equation (NLS) with confining harmonic potential $i{\partial}_{t}u+1/2\Delta u-1/2\left|x\right|\xb2u=-{\left|u\right|}^{4/(d-2)}u,x\in {\mathbb{R}}^{d}$, is considered. By modifying a variational technique, we shall give a sufficient condition under which the corresponding solution blows up.

This paper is concerned with the initial boundary value problem for a nonlocal p-Laplacian evolution equation with critical initial energy. In the framework of the energy method, we construct an unstable set and establish its invariance. Finally, the finite time blow-up of solutions is derived by a combination of the unstable set and the concavity method.

In this paper, we consider the nonlinear Kirchhoff-type equation $${u}_{tt}+M\left({\u2225{D}^{m}u\left(t\right)\u2225}_{2}^{2}\right){(-\Delta )}^{m}u+{\left|{u}_{t}\right|}^{q-2}{u}_{t}={\left|{u}_{t}\right|}^{p-2}u$$ with initial conditions and homogeneous boundary conditions. Under suitable conditions on the initial datum, we prove that the solution blows up in finite time.