Related papers: Global solutions of quasilinear wave equations

We consider a class of abstract quasilinear parabolic problems with lower--order terms exhibiting a prescribed singular structure. We prove well--posedness and Lipschitz continuity of associated semiflows. Moreover, we investigate global…

This paper studies the upper bound of the lifespan of classical solutions of the initial value problems for one dimensional wave equations with quasilinear terms of space-, or time-derivatives of the unknown function. The results are same…

We prove that every solution of the focusing energy-critical wave equation with the compactness property is global. We also give similar results for supercritical wave and Schr\"odinger equations.

It is shown that if the system of the Euler equations has a special global in time smooth solution with the linear profile of velocity, then another solutions with Cauchy data, close in the Sobolev norm to the initial data of the given…

We prove the existence of the modified wave operators for a scalar quasilinear wave equation satisfying the weak null condition. This is accomplished in three steps. First, we derive a new reduced asymptotic system for the quasilinear wave…

We study the global existence of Einstein-Maxwell(EM) equations on $\mathbb{R}^4$. We use the method, which relies on wave and Lorentzian gauge conditions, to obtain some exquisite estimates. Our main conclusion is that if the initial data…

This article represents the second installment of a series of papers concerned with low regularity solutions for the water wave equations in two space dimensions. Our focus here is on global solutions for small and localized data. Such…

We consider the gravity water waves system in the case of a one dimensional interface, for sufficiently smooth and localized initial data, and prove global existence of small solutions. This improves the almost global existence result of Wu…

We prove the global strong solvability of a quasilinear initial-boundary value problem with fractional time derivative of order less than one. Such problems arise in mathematical physics in the context of anomalous diffusion and the…

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, ...,…

We prove general a priori estimates of solutions of a class of quasilinear elliptic system on Carnot groups. As a consequence, we obtain several non existence theorems. The results are new even in the Euclidean setting.

We show that the spherically symmetric Einstein-scalar-field equations for wave-like decaying initial data at null infinity have unique local solutions and unique global solutions for small initial data. We also generalize Christodoulou's…

We study global existence and decay estimates for quasilinear wave equations with dissipative terms in the Sobolev space $H^L \times H^{L-1}$, where $L \geq [d/2]+3$. The linear dissipative terms depend on space variable coefficient, and…

We prove short time existence, uniqueness and continuous dependence on the initial data of smooth solutions of quasilinear locally parabolic equations of arbitrary even order on closed manifolds.

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…

We prove that the classical hyperbolic Kirchhoff equation admits global-in-time solutions for some classes of initial data in the energy space. We also show that there are enough such solutions so that every initial datum in the energy…

We prove that the subquartic wave equation on the three dimensional ball $\Theta$, with Dirichlet boundary conditions admits global strong solutions for a large set of random supercritical initial data in $\cap_{s<1/2} H^s(\Theta)$. We…

For the 3-D quadratic quasilinear wave equations in exterior domains with Dirichlet or Neumann boundary conditions, the global existence or the maximal existence time of small data smooth solutions have been established in the past.…

This work extends the previous work by the first author [arXiv:2409.02516] and [Math. Ann. 393 (2025), 317-363], analyzing the long-term behavior of solutions to a broader class of quasilinear wave equations with parameter…

Motivated by \cite{CG10,CZ6}, we prove the global existence of solutions to the two-dimensional isentropic compressible Navier-Stokes equations with smooth initial data which are slowly varying in one direction and with initial density…