Third order homogeneous weakly hyperbolic equations with nonanalytic coefficients

We consider the Cauchy problem for homogeneous linear third order weakly hyperbolic equations with time depending coefficients. We study the relation between the regularity of the coefficients and the Gevrey class in which the Cauchy problem is well-posed.     

Quantitative uniqueness for elliptic equations with singular lower order terms

We use a Carleman type inequality of Koch and Tataru to obtain quantitative estimates of unique continuation for solutions of second-order elliptic equations with singular lower order terms. First we prove a three sphere inequality and then describe two methods of propagation of smallness from sets of positive measure.     

Weakly hyperbolic equations with Lipschitz or Hölder continuous coefficients with respect to time

We consider second order weakly hyerbolic equations, with Lipschitz or Hölder continuous coefficients with respect to time, concerning the well posedness of the Cauchy problem and the propagation of the singularities in the framework of Gevrey classes.     

The Cauchy problem for nonstrictly second order hyperbolic equations with nonregular coefficients

In this paper we consider the Cauchy problem for the equation , where the matrix {a _jk(x)} is non-negative, and the first derivatives of the coefficients have a singularity of orderq≥3 att=T>0; under these assumptions, the Cauchy problem is well-posed in all Gevrey classes of indexs<q/(q−1).     

Some noncoercive parabolic equations with lower order terms in divergence form

This paper deals with existence and regularity results for the problem $ \cases{u_t-\mathrm{div}(a(x,t,u )\nabla u)=-\mathrm{div}(u\,E) \qquad in \Omega\times (0,T),\cr u=0 \qquad on \partial \Omega\times (0,T), \cr u (0)= u_0 \qquad in \Omega ,\cr} $ under various assumptions on E and $ u_0 $. The main difculty in studying this problem is due to the presence of the term div(uE), which makes the differential operator non coercive on the "energy space" $ L^2 (0, T; H_0^1 (\Omega)) $.AMS Subject Classification: 35K10, 35K15, 35K65.     

Weakly hyperbolic systems with Hölder continuous coefficients

We study the Cauchy Problem for a hyperbolic system with multiple characteristics and non-smooth coefficients depending on time. We prove in particular that, if the leading coefficients are α-Hölder continuous, and the system has size m?3, then the Problem is well posed in each Gevrey class of exponent s<1+α/m.     

On second order weakly hyperbolic equations with oscillating coefficients and regularity loss of the solutions

Regularity of the solution for the wave equation with constant propagation speed is conserved with respect to time, but such a property is not true in general if the propagation speed is variable with respect to time. The main purpose of this paper is to describe the order of regularity loss of the solution due to the variable coefficient by the following four properties of the coefficient: “smoothness”, “oscillations”, “degeneration” and “stabilization”. Actually, we prove the Gevrey and C^∞ well‐posedness for the wave equations with degenerate coefficients taking into account the interactions of these four properties. Moreover, we prove optimality of these results by constructing some examples (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Upper and lower solutions method for impulsive partial hyperbolic differential equations with fractional order

In this paper we investigate the existence of solutions for a class of initial value problems for impulsive partial hyperbolic differential equations involving the Caputo fractional derivative by using the lower and upper solutions method combined with Schauder’s fixed point theorem.     

High order difference schemes for the system of two space second order nonlinear hyperbolic equations with variable coefficients

In this paper, we develop implicit difference schemes of O(k⁴ + k²h² + h⁴), where k > 0, h > 0 are grid sizes in time and space coordinates, respectively, for solving the system of two space dimensional second order nonlinear hyperbolic partial differential equations with variable coefficients having mixed derivatives subject to appropriate initial and boundary conditions. The proposed difference method for the scalar equation is applied for the solution of wave equation in polar coordinates to obtain three level conditionally stable ADI method of O(k⁴ + k²h² + h⁴). Some physical nonlinear problems are provided to demonstrate the accuracy of the implementation.     

Boundary blow-up solutions of p-Laplacian elliptic equations with lower order terms

We study the existence, uniqueness, and asymptotic behavior of blow-up solutions for a general quasilinear elliptic equation of the type −Δ _p u = a(x)u ^m −b(x)f(u) with p >  1 and 0 <  m <  p−1. The main technical tool is a new comparison principle that enables us to extend arguments for semilinear equations to quasilinear ones. Indeed, this paper is an attempt to generalize all available results for the semilinear case with p =  2 to the quasilinear case with p >  1.