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Register a new userThe interplay of critical regularity of nonlinearities in a weakly coupled system of semi-linear damped wave equations
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We would like to study a weakly coupled system of semi-linear classical damped wave equations with moduli of continuity in nonlinearities whose powers belong to the critical curve in the $p-q$ plane. The main goal of this paper is to find out the sharp conditions of these moduli of continuity which classify between global (in time) existence of small data solutions and finite time blow-up of solutions.
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In this paper we obtain higher order asymptotic profilles of solutions to the Cauchy problem of the linear damped wave equation in $textbf{R}^n$ begin{equation*} u_{tt}-Delta u+u_t=0, qquad u(0,x)=u_0(x), quad u_t(0,x)=u_1(x), end{equation*} where $nintextbf{N}$ and $u_0$, $u_1in L^2(textbf{R}^n)$. Established hyperbolic part of asymptotic expansion seems to be new in the sense that the order of the expansion of the hyperbolic part depends on the spatial dimension.
We consider a coupled Wave-Klein-Gordon system in 3D, and prove global regularity and modified scattering for small and smooth initial data with suitable decay at infinity. This system was derived by Wang and LeFloch-Ma as a simplified model for the global nonlinear stability of the Minkowski space-time for self-gravitating massive fields.
The exact distributed controllability of the semilinear wave equation $partial_{tt}y-Delta y + g(y)=f ,1_{omega}$ posed over multi-dimensional and bounded domains, assuming that $gin C^1(mathbb{R})$ satisfies the growth condition $limsup_{rto infty} g(r)/(vert rvert ln^{1/2}vert rvert)=0$ has been obtained by Fu, Yong and Zhang in 2007. The proof based on a non constructive Leray-Schauder fixed point theorem makes use of precise estimates of the observability constant for a linearized wave equation. Assuming that $g^prime$ does not grow faster than $beta ln^{1/2}vert rvert$ at infinity for $beta>0$ small enough and that $g^prime$ is uniformly Holder continuous on $mathbb{R}$ with exponent $sin (0,1]$, we design a constructive proof yielding an explicit sequence converging to a controlled solution for the semilinear equation, at least with order $1+s$ after a finite number of iterations.
We consider positive solutions of $Delta u-mu u+Ku^{frac{n+2}{n-2}}=0$ on $B_1$ ($nge 5$) where $mu $ and $K>0$ are smooth functions on $B_1$. If $K$ is very sub-harmonic at each critical point of $K$ in $B_{2/3}$ and the maximum of $u$ in $bar B_{1/3}$ is comparable to its maximum over $bar B_1$, then all positive solutions are uniformly bounded on $bar B_{1/3}$. As an application, a priori estimate for solutions of equations defined on $mathbb S^n$ is derived.
The paper gives a comprehensive study of infinite-energy solutions and their long-time behavior for semi-linear weakly damped wave equations in $mathbb{R}^3$ with quintic nonlinearities. This study includes global well-posedness of the so-called Shatah-Struwe solutions, their dissipativity, the existence of a locally compact global attractors (in the uniformly local phase spaces) and their extra regularity.
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