Blow-up results for some second order hyperbolic inequalities with a nonlinear term with respect to the velocity

We give sufficient conditions on the initial data so that a semilinear wave inequality blows-up in finite time. Our method is based on the study of an associated second order differential inequality. The same method is applied to some semilinear systems of mixed type.Comment: 13 pages, 2 figure

A test function method for evolution equations with fractional powers of the Laplace operator

In this paper, we discuss a test function method to obtain nonexistence of global-in-time solutions for higher order evolution equations with fractional derivatives and a power nonlinearity, under a sign condition on the initial data. In order to deal with fractional powers of the Laplace operator, we introduce a suitable test function and a suitable class of weak solutions. The optimality of the nonexistence result provided is guaranteed by both scaling arguments and counterexamples. In particular, our manuscript provides the counterpart of nonexistence for several recent results of global existence of small data solutions to the following problem: $$\begin{cases} u_{tt} + (-\Delta)^{\theta}u_t + (-\Delta)^{\sigma} u = f(u,u_t),& t>0, \ x\in\mathbb R^n,\\ u(0,x)=u_0(x), \ u_t(0,x)=u_1(x) \end{cases}$$ with $f=|u|^p$ or $f=|u_t|^p$, where $\theta\geq0$ and $\sigma>0$ are fractional powers.Comment: 24pages, no figur

Blow-up scaling and global behaviour of solutions of the bi-Laplace equation via pencil operators

As the main problem, the bi-Laplace equation $\Delta^2u=0 (\Delta=D_x^2+D_y^2)$ in a bounded domain \Omega \subset \re^2, with inhomogeneous Dirichlet or Navier-type conditions on the smooth boundary $\partial \Omega$ is considered. In addition, there is a finite collection of curves \Gamma = \Gamma_1\cup...\cup\Gamma_m \subset \Omega, \quad \mbox{on which we assume homogeneous Dirichlet} \quad u=0, focusing at the origin $0 \in \Omega$ (the analysis would be similar for any other point). This makes the above elliptic problem overdetermined. Possible types of the behaviour of solution $u(x,y)$ at the tip $0$ of such admissible multiple cracks, being a singularity point, are described, on the basis of blow-up scaling techniques and spectral theory of pencils of non self-adjoint operators. Typical types of admissible cracks are shown to be governed by nodal sets of a countable family of harmonic polynomials, which are now represented as pencil eigenfunctions, instead of their classical representation via a standard Sturm--Liouville problem. Eventually, for a fixed admissible crack formation at the origin, this allows us to describe all boundary data, which can generate such a blow-up crack structure. In particular, it is shown how the co-dimension of this data set increases with the number of asymptotically straight-line cracks focusing at 0

A blow-up result for a semilinear wave equation with scale-invariant damping and mass and nonlinearity of derivative type

In this note, we prove blow-up results for semilinear wave models with damping and mass in the scale-invariant case and with nonlinear terms of derivative type. We consider the single equation and the weakly coupled system. In the first case we get a blow-up result for exponents below a certain shift of the Glassey exponent. For the weakly coupled system we find as critical curve a shift of the corresponding curve for the weakly coupled system of semilinear wave equations with the same kind of nonlinearities. Our approach follows the one for the respective classical wave equation by Zhou Yi. In particular, an explicit integral representation formula for a solution of the corresponding linear scale-invariant wave equation, which is derived by using Yagdjian's integral transform approach, is employed in the blow-up argument. While in the case of the single equation we may use a comparison argument, for the weakly coupled system an iteration argument is applied

Small data blow-up for a system of nonlinear Schr\"odinger equations

We give examples of small data blow-up for a three-component system of quadratic nonlinear Schr\"odinger equations in one space dimension. Our construction of the blowing-up solution is based on the Hopf-Cole transformation, which allows us to reduce the problem to getting suitable growth estimates for a solution to the transformed system. Amplification in the reduced system is shown to have a close connection with the mass resonance.Comment: 14 pages, to appear in J.Math.Anal.App

Degeneracy in finite time of 1D quasilinear wave equations II

We consider the large time behavior of solutions to the following nonlinear wave equation: $\partial_{t}^2 u = c(u)^{2}\partial^2_x u + \lambda c(u)c'(u)(\partial_x u)^2$ with the parameter $\lambda \in [0,2]$. If $c(u(0,x))$ is bounded away from a positive constant, we can construct a local solution for smooth initial data. However, if $c(\cdot )$ has a zero point, then $c(u(t,x))$ can be going to zero in finite time. When $c(u(t,x))$ is going to 0, the equation degenerates. We give a sufficient condition that the equation with $0\leq \lambda < 2$ degenerates in finite time.Comment: 14 pages, 1 figure. Some typos are fixed. The proof in the case u_1 does not belong to L^1 adde

Well-posedness for Hardy-H\'enon parabolic equations with fractional Brownian noise

We study the Hardy-H\'enon parabolic equations on $\mathbb{R}^{N}$ ($N=2, 3$) under the effect of an additive fractional Brownian noise with Hurst parameter $H>\max\left(1/2, N/4\right).$ We show local existence and uniqueness of a mid $L^{q}$-solution under suitable assumptions on $q$

Global weak solutions for generalized SQG in bounded domains

We prove the existence of global $L^2$ weak solutions for a family of generalized inviscid surface-quasi geostrophic (SQG) equations in bounded domains of the plane. In these equations, the active scalar is transported by a velocity field which is determined by the scalar through a more singular nonlocal operator compared to the SQG equation. The result is obtained by establishing appropriate commutator representations for the weak formulation together with good bounds for them in bounded domains.Comment: 15 page

Semi-linear structural damped waves

We study the global existence of small data solutions for Cauchy problem for the semi-linear structural damped wave equation with source term.Comment: 24 Page

Local Hadamard well-posedness and blow-up for reaction-diffusion equations with non-linear dynamical boundary conditions

The paper deals with local well-posedness, global existence and blow-up results for reaction--diffusion equations coupled with nonlinear dynamical boundary conditions