If Ω is a Lip(1,1/2) domain, μ a doubling measure on ${\partial }_p\Omega ,\partial /\partial t-L_i$, i = 0,1, are two parabolic-type operators with coefficients bounded and measurable, 2 ≤ q < ∞, then the associated measures ${\omega }_0$, ${\omega }_1$ have the property that ${\omega }_0\in B^q\left(\mu \right)$ implies ${\omega }_1$ is absolutely continuous with respect to ${\omega }_0$ whenever a certain Carleson-type condition holds on the difference function of the coefficients of $L_1$ and $L_0$. Also ${\omega }_0\in B^q\left(\mu \right)$ implies ${\omega }_1\in B^q\left(\mu \right)$ whenever both measures are center-doubling measures. This is B. Dahlberg’s result for elliptic measures extended...

We give a sufficient condition on the coefficients of a class of infinite horizon backward doubly stochastic differential equations (BDSDES), under which the infinite horizon BDSDES have a unique solution for any given square integrable terminal values. We also show continuous dependence theorem and convergence theorem for this kind of equations.

In this paper, we show the backward uniqueness in time of solutions to nonlinear integro-differential systems with Neumann or Dirichlet boundary conditions. We also discuss reasonable physical interpretations for our conclusions.

We establish the existence, uniqueness and main properties of the fundamental solutions for the fractional porous medium equation introduced in [51]. They are self-similar functions of the form $u(x,t)=t^{–\alpha }f\left(\right|x|t^{–\beta })$ with suitable $$ and $\beta$. As a main application of this construction, we prove that the asymptotic behaviour of general solutions is represented by such special solutions. Very singular solutions are also constructed. Among other interesting qualitative properties of the equation we prove an Aleksandrov reflection...

While alternans in a single cardiac cell appears through a simple period-doubling bifurcation, in extended tissue the exact nature of the bifurcation is unclear. In particular, the phase of alternans can exhibit wave-like spatial dependence, either stationary or travelling, which is known as discordant alternans. We study these phenomena in simple cardiac models through a modulation equation proposed by Echebarria-Karma. As shown in our previous paper, the zero solution of their equation may lose...

We investigate critical exponents for blow-up of nonnegative solutions to a class of parabolic inequalities. The proofs make use of a priori estimates of solutions combined with a simple scaling argument.

In this paper, we consider the nonlinear Kirchhoff-type equation $$u_{tt}+M\left({∥D^{m}u\left(t\right)∥}_2^2\right){(-\Delta )}^{m}u+{\left|u_t\right|}^{q-2}{u}_t={\left|u_t\right|}^{p-2}u$$ with initial conditions and homogeneous boundary conditions. Under suitable conditions on the initial datum, we prove that the solution blows up in finite time.

A class of nonlinear viscous transport equations describing aggregation phenomena in biology is considered. General conditions on an interaction potential are obtained which lead either to the existence or to the nonexistence of global-in-time solutions.

In this paper we discuss the approximate reconstruction of inhomogeneities of small volume. The data used for the reconstruction consist of boundary integrals of the (observed) electromagnetic fields. The numerical algorithms discussed are based on highly accurate asymptotic formulae for the electromagnetic fields in the presence of small volume inhomogeneities.

In this paper we discuss the approximate reconstruction of inhomogeneities of small volume. The data used for the reconstruction consist of boundary integrals of the (observed) electromagnetic fields. The numerical algorithms discussed are based on highly accurate asymptotic formulae for the electromagnetic fields in the presence of small volume inhomogeneities.

Let $D$ be either the unit ball $B_1\left(0\right)$ or the half ball $B_1^{+}\left(0\right),$ let $f$ be a strictly positive and continuous function, and let $u$ and $\Omega \subset D$ solve the following overdetermined problem:$$\Delta u\left(x\right)={\chi }_{{}_{\Omega }}\left(x\right)f\left(x\right)\text{in}D,0\in \partial \Omega ,u=\left|\nabla u\right|=0\text{in}{\Omega }^c,$$where ${\chi }_{{}_{\Omega }}$ denotes the characteristic function of $\Omega ,$${\Omega }^c$ denotes the set $D\setminus \Omega ,$ and the equation is satisfied in the...

We develop a new method for proving the existence of a boundary trace, in the class of Borel measures, of nonnegative solutions of $-\Delta u+g(x,u)\ge 0$ in a smooth domain $\Omega$ under very general assumptions on $g$. This new definition which extends the previous notions of boundary trace is based upon a sweeping technique by solutions of Dirichlet problems with measure boundary data. We also prove a boundary pointwise blow-up estimate of any solution of such inequalities in terms of the Poisson kernel. If the nonlinearity...