Related papers: Global smooth solutions to 2D semilinear wave equa…
We prove an extended lifespan result for the full gravity-capillary water waves system with a $2$ dimensional periodic interface: for initial data of sufficiently small size $\varepsilon$, smooth solutions exist up to times of the order of…
The classical system of shallow-water (Saint--Venant) equations describes long surface waves in an inviscid incompressible fluid of a variable depth. Although shock waves are expected in this quasilinear hyperbolic system for a wide class…
Our interest itself of this paper is strongly inspired from an open problem in the paper [1] published by D'Abbicco. In this article, we would like to study the Cauchy problem for a weakly coupled system of semi-linear structurally damped…
For the $2$-D semilinear wave equation with scale-invariant damping $\partial_t^2u-\Delta u+\frac{\mu}{t}\partial_tu=|u|^p$, where $t\ge 1$ and $p>1$, in the paper [T. Imai, M. Kato, H. Takamura, K. Wakasa, The lifespan of solutions of…
We prove almost global well-posedness for quasilinear strongly coupled wave-Klein-Gordon systems with small and localized data in two space dimensions. We assume only mild decay on the data at infinity as well as minimal regularity. We…
We prove small-data global existence to semi-linear wave equations on hyperbolic space of dimension greater than or equal to three, for nonlinearities that have the form of a sufficiently high integer power of the solution. We also prove…
We show that a general class of quasilinear wave equations have global solutions for small initial data as we conjectured in an earlier paper.
Coupled wave equations are popular tool for investigating longitudinal dynamical effects in semiconductor lasers, for example, sensitivity to delayed optical feedback. We study a model that consists of a hyperbolic linear system of partial…
In this paper we study global nonlinear stability for a system of semilinear wave and Klein-Gordon equations with quadratic nonlinearities. We consider nonlinearities of the type of wave-Klein-Gordon interactions where there are no…
In connection with the weak null condition, Alinhac introduced a sufficient condition for global existence of small amplitude solutions to systems of semilinear wave equations in three space dimensions. We introduce a slightly weaker…
We provide a proof of global existence of solutions to quasilinear wave equations satisfying the null condition in certain exterior domains. In particular, our proof does not require estimation of the fundamental solution for the free wave…
We investigate the semilinear wave equation with potential on weighted graphs. We establish sufficient conditions for the nonexistence of global-in-time solutions. Both nonnegative and sign-changing solutions are considered. In particular,…
In to previous papers by the authors, classes of initial data to the three dimensional, incompressible Navier-Stokes equations were presented, generating a global smooth solution although the norm of the initial data may be chosen…
A weak formulation for the so-called "semilinear strongly damped wave equation with constraint" is introduced and a corresponding notion of solution is defined. The main idea in this approach consists in the use of duality techniques in…
We generalize the pointwise decay estimates for large data solutions of the defocusing semilinear wave equations which we obtained earlier under restriction to spherical symmetry. Without the symmetry the conformal transformation we use…
By introducing new weighted vector fields as multipliers, we derive quantitative pointwise estimates for solutions of defocusing semilinear wave equation in $\mathbb{R}^{1+3}$ with pure power nonlinearity for all $1<p\leq 2$. Consequently,…
This paper studies initial value problems for semilinear wave equations with spatial weights in one space dimension. The lifespan estimates of classical solutions for compactly supported data are established in all the cases of polynomial…
The paper is devoted to proving an existence and uniqueness result for generalized solutions to semilinear wave equations with a small nonlinearity in space dimensions 1, 2, 3. The setting is the one of Colombeau algebras of generalized…
In this paper, we are concerned with the 3-D quasilinear wave equation $ \ds\sum_{i,j=0}^3g^{ij}(u, \p u)\p_{ij}^2u$ $=0$ with $(u(0,x), \p_tu(0,x))=(\ve u_0(x), \ve u_1(x))$, where $x_0=t$, $x=(x_1, x_2, x_3)$, $\p=(\p_0, \p_1, ...,…
Global existence for small data Cauchy problem of semilinear wave equations with scaling invariant damping in 3-D is established in this work, assuming that the data are radial and the constant in front of the damping belongs to $[1.5, 2)$.…