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We consider the Cauchy problem for the nonlinear wave equation $u_{tt} - Delta_x u +q(t, x) u + u^3 = 0$ with smooth potential $q(t, x) geq 0$ having compact support with respect to $x$. The linear equation without the nonlinear term $u^3$ and potential periodic in $t$ may have solutions with exponentially increasing as $ t to infty$ norm $H^1({mathbb R}^3_x)$. In [2] it was established that adding the nonlinear term $u^3$ the $H^1({mathbb R}^3_x)$ norm of the solution is polynomially bounded for every choice of $q$. In this paper we show that $H^k({mathbb R}^3_x)$ norm of this global solution is also polynomially bounded. To prove this we apply a different argument based on the analysis of a sequence ${Y_k(ntau_k)}_{n = 0}^{infty}$ with suitably defined energy norm $Y_k(t)$ and $0 < tau_k <1.$
An algebraic lower bound on the energy decay for solutions of the advection-diffusion equation in $mathbb{R}^d$ with $d=2,3$ is derived using the Fourier splitting method. Motivated by a conjecture on mixing of passive scalars in fluids, a lower boun
The Cauchy problem of the modified nonlinear Schr{o}dinger (mNLS) equation with the finite density type initial data is investigated via $overline{partial}$ steepest descent method. In the soliton region of space-time $x/tin(5,7)$, the long-time asym
In this paper, we are going to investigate Cauchy problem for nonlocal nonlinear Schrodinger equation with the initial potential $q_0(x)$ in weighted sobolev space $H^{1,1}(mathbb{R})$, begin{align*} iq_t(x,t)&+q_{xx}(x,t)+2sigma q^2(x,t)bar q(-x,t)=
In this work, the $overline{partial}$ steepest descent method is employed to investigate the soliton resolution for the Hirota equation with the initial value belong to weighted Sobolev space $H^{1,1}(mathbb{R})={fin L^{2}(mathbb{R}): f,xfin L^{2}(ma
In this paper we prove the existence of vortices, namely standing waves with non null angular momentum, for the nonlinear Klein-Gordon equation in dimension $Ngeq 3$. We show with variational methods that the existence of these kind of solutions, tha