Nonlinear wave equation with potential

Nonlinear Dissipative Wave Equations with Potential

We study the long time behavior of solutions of the wave equation with absorbtion |u|p−1u and variable damping V (x)ut, where V (x) ∼ V0|x|−α for large |x|, V0 > 0. For α ∈ [0, 1) and 1 < p < (n + 2)/(n − 2) we establish decay estimates of the energy, L2 and Lp+1 norm of solutions. We find three different regimes of decay of solutions depending on the exponent of the absorbtion term. We show th...

Existence of Global Solution of a Nonlinear Wave Equation with Short-range Potential

The existence of a global solution for the Cauchy problem for the nonlinear wave equation (1) (∂tt −∆)u = F (u′, u′′) , t ≥ 0, x ∈ R , is established by S. Klainerman [6] under a suitable algebraic condition called the null condition on the quadratic nonlinearity in F (u′, u′′). The global or “almost global” solution of the corresponding mixed problem has been studied by Y. Shibata and Y. Tsuts...

Nonlinear Wave Equation in Special Coordinates

Some classical types of nonlinear periodic wave motion are studied in special coordinates. In the case of cylinder coordinates, the usual perturbation techniques leads to the overdetermined systems of linear algebraic equations for unknown coefficients whose compatibility is key step of the investigation. Their solutions give solutions to the nonlinear wave equation which are periodic in time a...

Forced Vibrations for a Nonlinear Wave Equation

Nonlinear wave evolution equation for critical layers.

Recent studies of the evolution of weakly nonlinear long waves in shear flows have revealed that when the wave field contains a critical layer, a new nonlinear wave equation is needed to describe the wave evolution. This equation is of the same type as the well-known Korteweg-de Vries equation but has a more complicated nonlinear structure. Our main interest is in the steady solitary wave solut...