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Register a new userA note on a strongly damped wave equation with fast growing nonlinearities
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A strongly damped wave equation including the displacement depending nonlinear damping term and nonlinear interaction function is considered. The main aim of the note is to show that under the standard dissipativity restrictions on the nonlinearities involved the initial boundary value problem for the considered equation is globally well-posed in the class of sufficiently regular solutions and the semigroup generated by the problem possesses a global attractor in the corresponding phase space. These results are obtained for the nonlinearities of an arbitrary polynomial growth and without the assumption that the considered problem has a global Lyapunov function.
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A weak formulation for the so-called semilinear strongly damped wave equation with constraint is introduced and a corresponding notion of solution is defined. The main idea in this approach consists in the use of duality techniques in Sobolev-Bochner spaces, aimed at providing a suitable relaxation of the constraint term. A global in time existence result is proved under the natural condition that the initial data have finite physical energy.
We present a new method of investigating the so-called quasi-linear strongly damped wave equations $$ partial_t^2u-gammapartial_tDelta_x u-Delta_x u+f(u)= abla_xcdot phi( abla_x u)+g $$ in bounded 3D domains. This method allows us to establish the existence and uniqueness of energy solutions in the case where the growth exponent of the non-linearity $phi$ is less than 6 and $f$ may have arbitrary polynomial growth rate. Moreover, the existence of a finite-dimensional global and exponential attractors for the solution semigroup associated with that equation and their additional regularity are also established. In a particular case $phiequiv0$ which corresponds to the so-called semi-linear strongly damped wave equation, our result allows to remove the long-standing growth restriction $|f(u)|leq C(1+ |u|^5)$.
We concern the blow up problem to the scale invariant damping wave equations with sub-Strauss exponent. This problem has been studied by Lai, Takamura and Wakasa (cite{Lai17}) and Ikeda and Sobajima cite{Ikedapre} recently. In present paper, we extend the blowup exponent from $p_F(n)leq p<p_S(n+2mu)$ to $1<p<p_S(n+mu)$ without small restriction on $mu$. Moreover, the upper bound of lifespan is derived with uniform estimate $T(varepsilon)leq Cvarepsilon^{-2p(p-1)/gamma(p,n+2mu)}$. This result extends the blowup result of semilinear wave equation and shows the wave-like behavior of scale invariant damping wave equations solution even with large $mu>1$.
This paper is devoted to the lifespan of solutions to a damped fourth-order wave equation with logarithmic nonlinearity $$u_{tt}+Delta^2u-Delta u-omegaDelta u_t+alpha(t)u_t=|u|^{p-2}uln|u|.$$ Finite time blow-up criteria for solutions at both lower and high initial energy levels are established, and an upper bound for the blow-up time is given for each case. Moreover, by constructing a new auxiliary functional and making full use of the strong damping term, a lower bound for the blow-up time is also derived.
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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