Compactness for the weighted Hardy operator in variable exponent spaces

In this paper, we prove a necessary and sufficiency condition for the weighted Hardy operator

$H_{\upsilon ,\omega }f(x)=\upsilon (x)\underset{0}{\overset{x}{\int }}f(t)\omega (t)\mathrm{d}t$

to be compactly acting from $L^{p(?)}(0,\infty )$ to $L^{q(?)}(0,\infty )$.     

Estimating the division rate and kernel in the fragmentation equation

We consider the fragmentation equation

$\frac{?f}{?t}(t,x)=?B(x)f(t,x)+\underset{y=x}{\overset{y=\infty }{\int }}k(y,x)B(y)f(t,y)dy,$

and address the question of estimating the fragmentation parameters – i.e. the division rate $B(x)$ and the fragmentation kernel $k(y,x)$ – from measurements of the size distribution $f(t,?)$ at various times. This is a natural question for any application where the sizes of the particles are measured experimentally whereas the fragmentation rates are unknown, see for instance Xue and Radford (2013) [26] for amyloid fibril breakage. Under the assumption of a polynomial division rate $B(x)=\alpha x^{\gamma }$ and a self-similar fragmentation kernel $k(y,x)=\frac{1}{y}{k}_0(\frac{x}{y})$, we use the asymptotic behavior proved in Escobedo et al. (2004) [11] to obtain uniqueness of the triplet $(\alpha ,\gamma ,k_0)$ and a representation formula for $k_0$. To invert this formula, one of the delicate points is to prove that the Mellin transform of the asymptotic profile never vanishes, what we do through the use of the Cauchy integral.     

Exponential stability for the wave model with localized memory in a past history framework

In this paper we discuss the asymptotic stability as well as the well-posedness of the damped wave equation posed on a bounded domain Ω of ${\mathbb{R}}^n,n\ge 2$,

$\rho (x)u_{tt}?\mathrm{\Delta }u+\underset{0}{\overset{\infty }{\int }}g(s)\text{div}[a(x)\mathrm{?}u(?,t?s)]ds+b(x)u_t=0,$

subject to a locally distributed viscoelastic effect driven by a nonnegative function $a(x)$ and supplemented with a frictional damping $b(x)\ge 0$ acting on a region A of Ω, where $a=0$ in A. Assuming that $\rho (x)$ is constant, considering that the well-known geometric control condition $(\omega ,T_0)$ holds and supposing that the relaxation function g is bounded by a function that decays exponentially to zero, we prove that the solutions to the corresponding partial viscoelastic model decay exponentially to zero, even in the absence of the frictional dissipative effect. In addition, in some suitable cases where the material density $\rho (x)$ is not constant, it is also possible to remove the frictional damping term $b(x)u_t$, that is, the localized viscoelastic damping is strong enough to assure that the system is exponentially stable. The semi-linear case is also considered.     

A characterization of well-posedness for abstract Cauchy problems with finite delay

Let A be a closed operator defined on a Banach space X and F be a bounded operator defined on a appropriate phase space. In this paper, we characterize the well-posedness of the first order abstract Cauchy problem with finite delay,

$\{\begin{array}{cc}{u}^{\prime }(t)=Au(t)+F{u}_t,\hfill & \hfill t>0;\hfill \\ u(0)=x;\hfill \\ u(t)=?(t),\hfill & \hfill ?r\le t<0\hfill \end{array},$

solely in terms of a strongly continuous one-parameter family ${\{G(t)\}}_{t\ge 0}$ of bounded linear operators that satisfy the functional equation

$G(t+s)x=G(t)G(s)x+\underset{?r}{\overset{0}{\int }}G(t+m)[SG(s+?)x](m)dm$

for all $t,s\ge 0,x\in X$. In case $F\equiv 0$ this property reduces to the characterization of well-posedness for the first order abstract Cauchy problem in terms of the functional equation that satisfy the $C_0$-semigroup generated by A.     

Weighted bilinear Hardy inequalities

We characterize the weights w, $w_1$, $w_2$ such that the weighted bilinear Hardy inequality$\left(\underset{a}{\overset{b}{\int }}\right(\underset{a}{\overset{x}{\int }}f{)}^q(\underset{a}{\overset{x}{\int }}g{)}^{q}w(x)dx{)}^{\frac{1}{q}}?C(\underset{a}{\overset{b}{\int }}{f}^{p_1}{w}_1{)}^{\frac{1}{p_1}}(\underset{a}{\overset{b}{\int }}{g}^{p_2}{w}_2{)}^{\frac{1}{p_2}}$ holds for all nonnegative functions f and g, with a positive constant C independent of f and g, for all possible values of q, $p_1$ and $p_2$ with $1<q,p_1,p_2<\infty$. We also characterize the good weights for the weighted bilinear n-dimensional Hardy inequality to hold.