We prove existence of weak solutions to doubly degenerate diffusion equations $$\dot{u}={\Delta }_p{u}^{m-1}+f(m,p\ge 2)$$ by Faedo-Galerkin approximation for general domains and general nonlinearities. More precisely, we discuss the equation in an abstract setting, which allows to choose function spaces corresponding to bounded or unbounded domains $\Omega \subset {ℝ}^n$ with Dirichlet or Neumann boundary conditions. The function $f$ can be an inhomogeneity or a nonlinearity involving terms of the form $f\left(u\right)$ or $div\left(F\right(u\left)\right)$. In the appendix, an introduction to weak differentiability...

We prove the existence of a renormalized solution to a class of doubly nonlinear parabolic systems.

We investigate the existence of renormalized solutions for some nonlinear parabolic problems associated to equations of the form ⎧$\partial (e^{\beta u}-1)/\partial t-{div\left(\right|\nabla u|}^{p-2}\nabla u)+div(c(x,t){\left|u\right|}^{s-1}u)+b(x,t){\left|\nabla u\right|}^r=f$ in Q = Ω×(0,T), ⎨ u(x,t) = 0 on ∂Ω ×(0,T), ⎩ $(e^{\beta u}-1)(x,0)=(e^{\beta u₀}-1)\left(x\right)$ in Ω. with s = (N+2)/(N+p) (p-1), $c(x,t)\in {\left(L^{\tau }\left(QT\right)\right)}^N$, τ = (N+p)/(p-1), r = (N(p-1) + p)/(N+2), $b(x,t)\in L^{N+2,1}\left(QT\right)$ and f ∈ L¹(Q).

We study the interaction of (slowly modulated) high frequency waves for multi-dimensional nonlinear Schrödinger equations with Gauge invariant power-law nonlinearities and nonlocal perturbations. The model includes the Davey-Stewartson system in its elliptic-elliptic and hyperbolic-elliptic variants. Our analysis reveals a new localization phenomenon for nonlocal perturbations in the high frequency regime and allows us to infer strong instability results on the Cauchy problem in negative order Sobolev...

We deal with a generalization of the Caginalp phase-field model associated with Neumann boundary conditions. We prove that the problem is well posed, before studying the long time behavior of solutions. We establish the existence of the global attractor, but also of exponential attractors. Finally, we study the spatial behavior of solutions in a semi-infinite cylinder, assuming that such solutions exist.

In this paper, the mixed initial-boundary value problem for inhomogeneous quasilinear strictly hyperbolic systems with nonlinear boundary conditions in the first quadrant $\left\{\right(t,x):t\ge 0,x\ge 0\}$ is investigated. Under the assumption that the right-hand side satisfies a matching condition and the system is strictly hyperbolic and weakly linearly degenerate, we obtain the global existence and uniqueness of a $C^1$ solution and its $L^1$ stability with certain small initial and boundary data.

We consider the Bresse system in bounded domain with delay terms in the internal feedbacks and prove the global existence of its solutions in Sobolev spaces by means of semigroup theory under a condition between the weight of the delay terms in the feedbacks and the weight of the terms without delay. Furthermore, we study the asymptotic behavior of solutions using multiplier method.

A viscoelastic Kirchhoff equation with Balakrishnan-Taylor damping is considered. Using integral inequalities and multiplier techniques we establish polynomial decay estimates for the energy of the problem. The results obtained in this paper extend previous results by Tatar and Zaraï [25].

This paper is concerned with the 3-D Cauchy problem for the compressible viscous fluid flow taking into account the radiation effect. For more general gases including ideal polytropic gas, we prove that there exists a unique smooth solutions in $[0,\infty )$, provided that the initial perturbations are small. Moreover, the time decay rates of the global solutions are obtained for higher-order spatial derivatives of density, velocity, temperature, and the radiative heat flux.

In this paper, we prove the existence of a global solution to an initial-boundary value problem for 1-D flows of the viscous heat-conducting radiative and reactive gases. The key point here is that the growth exponent of heat conductivity is allowed to be any nonnegative constant; in particular, constant heat conductivity is allowed.

In this paper, we will study global well-posedness for the cubic defocusing nonlinear Schrödinger equations on the compact Riemannian manifold without boundary, below the energy space, i.e. $s\<1$, under some bilinear Strichartz assumption. We will find some $\tilde{s}\<1$, such that the solution is global for $s\>\tilde{s}$.

This paper is devoted to the analysis of a one-dimensional model for phase transition phenomena in thermoviscoelastic materials. The corresponding parabolic-hyperbolic PDE system features a strongly nonlinear internal energy balance equation, governing the evolution of the absolute temperature $\vartheta$, an evolution equation for the phase change parameter $\chi$, including constraints on the phase variable, and a hyperbolic stress-strain relation for the displacement variable $𝐮$. The main novelty of the model...

We consider the initial-value problem for a nonlinear hyperbolic-parabolic system of three coupled partial differential equations of second order describing the process of thermodiffusion in a solid body (in one-dimensional space). We prove global (in time) existence and uniqueness of the solution to the initial-value problem for this nonlinear system. The global existence is proved using time decay estimates for the solution of the associated linearized problem. Next, we prove an energy estimate...

We prove small data global existence and scattering for quasilinear systems of Klein-Gordon equations with different speeds, in dimension three. As an application, we obtain a robust global stability result for the Euler-Maxwell equations for electrons.

This paper deals with a kind of hyperbolic boundary value problems with equivalued surface on a domain with thin layer. Existence and uniqueness of solutions are given, and the limit behavior of solutions is studied in this paper.

This paper considers the existence and uniqueness of the solution to the initial boundary value problem for a class of generalized Zakharov equations in $(2+1)$ dimensions, and proves the global existence of the solution to the problem by a priori integral estimates and the Galerkin method.

This article focuses on dynamic description of the collective pedestrian motion based on the kinetic model of Bhatnagar-Gross-Krook. The proposed mathematical model is based on a tendency of pedestrians to reach a state of equilibrium within a certain time of relaxation. An approximation of the Maxwellian function representing this equilibrium state is determined. A result of the existence and uniqueness of the discrete velocity model is demonstrated. Thus, the convergence of the solution to that...

We prove global stability results of DiPerna-Lionsrenormalized solutions for the initial boundary value problem associated to some kinetic equations, from which existence results classically follow. The (possibly nonlinear) boundary conditions are completely or partially diffuse, which includes the so-called Maxwell boundary conditions, and we prove that it is realized (it is not only a boundary inequality condition as it has been established in previous works). We are able to deal with Boltzmann,...