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### Open Access Journals

PROC

In this paper we show existence of finite energy solutions for the
Cauchy problem associated with a semilinear wave equation with
interior damping and supercritical source terms. The main contribution consists in dealing with super-supercritical source terms (terms of the order of $|u|^p$ with $p\geq 5$ in $n=3$ dimensions), an open and highly recognized problem in the literature on nonlinear wave equations.

EECT

We consider semilinear evolution equations of the form $a(t)\partial_{tt}u + b(t) \partial_t u + Lu = f(x,u)$ and $b(t) \partial_t u + Lu = f(x,u),$ with possibly unbounded $a(t)$ and possibly sign-changing damping coefficient $b(t)$, and determine precise conditions for which linear instability of the steady state solutions implies nonlinear instability. More specifically, we prove that linear instability with an eigenfunction of fixed sign gives rise to nonlinear instability by either exponential growth or finite-time blow-up. We then discuss a few examples to which our main theorem is immediately applicable, including evolution equations with supercritical and exponential nonlinearities.

keywords:
steady states.
,
variable coefficients
,
instability
,
Evolution equations
,
sign-changing damping

EECT

This note is an errata for the paper [2] which discusses regular solutions to wave equations with super-critical source terms.
The purpose of this note is to address the gap in the proof of

*uniqueness*of such solutions.
keywords:
nonlinear damping.
,
super-critical
,
critical exponent
,
regular solutions
,
Wave equation

EECT

We study regular solutions to wave equations with super-critical source terms, e.g., of exponent $p>5$ in 3D. Such sources have been a major challenge in the investigation of finite-energy ($H^1 \times L^2$) solutions to wave PDEs for many years. The wellposedness has been settled in part, but even the local existence, for instance, in 3 dimensions requires the relation $p\leq 6m/(m+1)$ between the exponents $p$ of the source and $m$ of the viscous damping.

We prove that smooth initial data ($H^2 \times H^1$) yields

We prove that smooth initial data ($H^2 \times H^1$) yields

*regular*solutions that do not depend on the above correlation. Local existence is demonstrated for any source exponent $p\geq 1$ and any monotone damping including feedbacks growing*exponentially or logarithmically at infinity, or with no damping at all*. The result holds in dimensions 3 and 4, and with some restrictions on $p$ in dimensions $n\geq 5$. Furthermore, if we assert the classical condition that the damping grows as fast as the source, then these regular solutions are global.
keywords:
critical exponent
,
regular solutions
,
nonlinear damping.
,
Wave equation
,
super-critical

EECT

In this paper we show local (and partially global) in time existence for the Westervelt equation with several
versions of nonlinear damping. This enables us to prove well-posedness with spatially varying $L_\infty$-coefficients,
which includes the situation of interface coupling between linear and nonlinear acoustics as well as between linear
elasticity and nonlinear acoustics, as relevant, e.g., in high intensity focused ultrasound (HIFU) applications.

DCDS-S

Under appropriate assumptions the energy of wave equations
with damping and variable coefficients $c(x)$$u_{t t}$-div$(b(x)\nabla
u)+a(x)u_t =h(x,t)$ has been shown to decay. Determining the decay rate for the higher order energies of the $k$th order spatial and time derivatives has been an open problem with the exception of some sparse results obtained for $k=1,2$. We establish the sharp gain in the decay rate for all higher order energies in terms of the first energy, and also obtain the sharp gain of decay rates for the $L^2$ norms of the higher order spatial derivatives. The results concern weighted (in time) and also pointwise (in time) energy decay estimates. We also obtain $L^\infty$ estimates for the solution $u$ in dimension $n=3$.
As an application we compute explicit decay rates for all energies which involve the dimension $n$
and the bounds for the coefficients $a(x)$ and $b(x)$ in the case $c (x)=1$ and $h(x,t)=0.$

EECT

This volume collects a number of contributions in the fields of partial differential equations and control theory, following the Special Session

For more information please click the “Full Text” above.

*Nonlinear PDEs and Control Theory with Applications*held at the 9th AIMS conference on Dynamical Systems, Differential Equations and Applications in Orlando, July 1--5, 2012.For more information please click the “Full Text” above.

keywords:

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