Existence of weak solutions to the Cauchy problem of a semilinear wave equation with supercritical interior source and damping
Lorena Bociu Petronela Radu
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.
keywords: damping and source terms wave equations energy identity weak solutions
Nonlinear instability of solutions in parabolic and hyperbolic diffusion
Stephen Pankavich Petronela Radu
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
Errata: Regular solutions of wave equations with super-critical sources and exponential-to-logarithmic damping
Lorena Bociu Petronela Radu Daniel Toundykov
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
Regular solutions of wave equations with super-critical sources and exponential-to-logarithmic damping
Lorena Bociu Petronela Radu Daniel Toundykov
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 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
Relaxation of regularity for the Westervelt equation by nonlinear damping with applications in acoustic-acoustic and elastic-acoustic coupling
Rainer Brunnhuber Barbara Kaltenbacher Petronela Radu
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.
keywords: nonlinear damping quasilinear wave equation local existence. Nonlinear acoustics
Higher order energy decay rates for damped wave equations with variable coefficients
Petronela Radu Grozdena Todorova Borislav Yordanov
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 $h(x,t)=0.$
keywords: decay rates higher order energy. hyperbolic diffusion linear dissipation wave equations with variable coefficients
Preface: Introduction to the Special Volume on Nonlinear PDEs and Control Theory with Applications
Lorena Bociu Barbara Kaltenbacher Petronela Radu
This volume collects a number of contributions in the fields of partial differential equations and control theory, following the Special Session 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.

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