Related papers: Large time behavior for the classical wave equatio…
The goal of the present paper is to study the viscoelastic wave equation with the time delay \[ |u_t|^\rho u_{tt}-\Delta u-\Delta u_{tt}+\int_0^tg(t-s)\Delta u(s)ds+\mu_1u_t(x,t)+\mu_2 u_t(x,t-\tau)=b|u|^{p-2}u\] under initial boundary…
We investigate the equation $(u_t + (f(u))_x)_x = f''(u) (u_x)^2/2$ where $f(u)$ is a given smooth function. Typically $f(u)= u^2/2$ or $u^3/3$. This equation models unidirectional and weakly nonlinear waves for the variational wave…
We consider the classical compressible Euler's Equations in three space dimensions with an arbitrary equation of state, and whose initial data corresponds to a constant state outside a sphere. Under suitable restriction on the size of the…
In this paper we study the large-time behavior of classical solutions to the two-species Vlasov-Maxwell-Boltzmann system in the whole space $\R^3$. The existence of global in time nearby Maxwellian solutions is known from [34] in 2006.…
A method based off of operator consideration for solving the time evolution of a wave function is developed. The method is applied to free space, constant force and harmonic oscillator potentials where general solutions are derived for the…
The Cauchy problem for two dimensional difference wave operators is considered with potentials and initial data supported in a bounded region. The large time asymptotic behavior of solutions is obtained. In contrast to the continuous case…
We obtain a decay estimate for solutions to the linear dispersive equation $iu_t-(-\Delta)^{1/4}u=0$ for $(t,x)\in\mathbb{R}\times\mathbb{R}$. This corresponds to a factorization of the linearized water wave equation…
The large time behavior of general solutions to a class of quasilinear diffusion equations with a weighted source term $$ \partial_tu=\Delta u^m+\varrho(x)u^p, \quad (x,t)\in\mathbb{R}^N\times(0,\infty), $$ with $m>1$, $1<p<m$ and suitable…
We study the Klein-Gordon equation with general interaction term, which may be linear or nonlinear, and space-time dependent. The initial data is general, large and non-radial. We prove that global solutions are asymptotically given by a…
We prove necessary optimality conditions of Euler-Lagrange type for generalized problems of the calculus of variations on time scales with a Lagrangian depending not only on the independent variable, an unknown function and its delta…
The model \[ \left\{ \begin{array}{l} u_{tt} = \big(\gamma(\Theta) u_{xt}\big)_x + au_{xx} - \big(f(\Theta)\big)_x, \\[1mm] \Theta_t = \Theta_{xx} + \gamma(\Theta) u_{xt}^2 - f(\Theta) u_{xt}, \end{array} \right. \] for thermoviscoelastic…
We study the large time behavior of solutions to the porous medium equation in nonhomogeneous media with critical singular density $$ |x|^{-2}\partial_{t}u=\Delta u^m, \quad \hbox{in} \ \real^N\times(0,\infty), $$ where $m>1$ and $N\geq3$.…
We study the large time behavior of nonnegative solutions of the Cauchy problem $u_t=\int J(x-y)(u(y,t)-u(x,t))\,dy-u^p$, $u(x,0)=u_0(x)\in L^\infty$, where $|x|^{\alpha}u_0(x)\to A>0$ as $|x|\to\infty$. One of our main goals is the study…
For the nonlinear wave equation $u_{tt} - c(u)\big(c(u) u_x\big)_x~=~0$, it is well known that solutions can develop singularities in finite time. For an open dense set of initial data, the present paper provides a detailed asymptotic…
We present both, theory and an algorithm for solving time-harmonic wave problems in a general setting. The time-harmonic solutions will be achieved by computing time-periodic solutions of the original wave equations. Thus, an exact…
In this paper, we prove global well-posedness with large initial data for the one-dimensional quasilinear wave equation $$ u_{tt}=c(u)^2u_{xx}, \qquad (t,x)\in (0,T)\times\R, $$ where \(c\) is a positive, bounded, monotonically increasing…
This manuscript considers the Jordan-Moore-Gibson-Thompson (JMGT) equation and its linearized equation with an additional weak damping term (proposed by [B. Kaltenbacher, \emph{Inverse Problems} (2025)] firstly) in the whole space…
We study the global existence of a unique strong solution and its large-time behavior of a two-phase fluid system consisting of the compressible isothermal Euler equations coupled with compressible isentropic Navier-Stokes equations through…
We study the large time behavior of solutions near a constant equilibrium to the compressible Euler-Maxwell system in $\r3$. We first refine a global existence theorem by assuming that the $H^3$ norm of the initial data is small, but the…
A new transient regime in the relaxation towards absolute equilibrium of the conservative and time-reversible 3-D Euler equation with high-wavenumber spectral truncation is characterized. Large-scale dissipative effects, caused by the…