Asymptotic parabolicity for strongly damped wave equations

For $S$ a positive selfadjoint operator on a Hilbert space, $$\frac{d^2u}{dt}(t) + 2 F(S)\frac{du}{dt}(t) + S^2u(t)=0$$ describes a class of wave equations with strong friction or damping if $F$ is a positive Borel function. Under suitable hypotheses, it is shown that $$u(t)=v(t)+ w(t)$$ where $v$ satisfies $$2F(S)\frac{dv}{dt}(t)+ S^2v(t)=0$$ and $$\frac{w(t)}{\|v(t)\|} \rightarrow 0, \; \text{as} \; t \rightarrow +\infty.$$ The required initial condition $v(0)$ is given in a canonical way in terms of $u(0)$, $u'(0)$

Higher order Ostrowski type inequalities over Euclidean domains

AbstractThe classical Ostrowski inequality for functions on intervals estimates the value of the function minus its average in terms of the maximum of its first derivative. This result is extended to functions on general domains using the L∞ norm of its nth partial derivatives. For radial functions on balls the inequality is sharp

Some Nonlinear Wave Equations with Acoustic Boundary Conditions

AbstractWe prove the existence and uniqueness of global solutions to the mixed problem for the Carrier equationutt−M∫Ωu2dxΔu+|u′t|αu′t=0with acoustic boundary conditions

Nonsymmetric elliptic operators with wentzell boundary conditions in general domains

We study nonsymmetric second order elliptic operators with Wentzell boundary conditions in general domains with sufficiently smooth boundary. The ambient space is a space of Lp-type, 1 ≤ p ≤ ∞. We prove the existence of analytic quasicontractive (C0)-semigroups generated by the closures of such operators, for any 1 ≤ p ≤ ∞. Moreover, we extend a previous result concerning the continuous dependence of these semigroups on the coefficients of the boundary condition. We also specify precisely the domains of the generators explicitly in the case of bounded domains and 1 ≤ p ≤ ∞, when all the ingredients of the problem, including the boundary of the domain, the coefficients, and the initial condition, are of class C∞

Instantaneous blowup and singular potentials on Heisenberg groups

In this paper we generalize the instantaneous blowup result from [3] and [15] to the heat equation perturbed by singular potentials on the Heisenberg group