Propagation of elastic waves is studied in a 1D medium containing N cracks modeled by nonlinear jump conditions. The case N = 1 is fully understood. When N > 1, the evolution equations are written as a system of nonlinear neutral delay differential equations, leading to a well-posed Cauchy problem. In the case N = 2, some mathematical results about the existence, uniqueness and attractivity of ...

In this paper, spectral properties of the Cauchy problem pantograph equation are examined. We establish boundaries parameter interval in which remains a Volterra problem.

We study the Cauchy problem for quasi-geostrophic equations in a unit ball of two-dimensional space with homogeneous Dirichlet boundary condition. show existence and uniqueness strong solution framework Besov spaces. also establish spectral localization technique commutator estimates.

Abstract. We consider the Cauchy problem We consider the Cauchy problem ut = uxx + f(u), x ∈ R, t > 0, u(x, 0) = u0(x), x ∈ R, where f is a locally Lipschitz function on R with f(0) = 0, and u0 is a nonnegative function in C0(R), the space of continuous functions with limits at ±∞ equal to 0. Assuming that the solution u is bounded, we study its large-time behavior from several points of view. ...

The explicit and exact solutions of the linear homogeneous difference equation with initial conditions (Cauchy problem) are constructed. The approach is quite general and relies on a novel and successful treatment of the linear recursion appropriately cast in matrix form. Our approach is exploited to solve the eigenvalues problem of a special set of non-Hermitian operators. A new class of gener...

We study asymptotic behavior of global positive solutions of the Cauchy problem for the semilinear parabolic equation ut = ∆u + up in RN , where p > 1 + 2/N , p(N − 2) ≤ N + 2. The initial data are of the form u(x, 0) = αφ(x), where φ is a fixed function with suitable decay at |x| = ∞ and α > 0 is a parameter. There exists a threshold parameter α∗ such that the solution exists globally if and o...

The probabilistic approach is used for constructing special layer methods to solve the Cauchy problem for semilinear parabolic equations with small parameter. Despite their probabilistic nature these methods are nevertheless deterministic. The algorithms are tested by simulating the Burgers equation with small viscosity and the generalized KPP-equation with a small parameter.

Derivatives of a solution of an ODE Cauchy problem can be computed inductively using the Faà di Bruno formula. In this paper, we exhibit a noninductive formula for these derivatives. At the heart of this formula is a combinatorial problem, which is solved in this paper. We also give a more tractable form of the Magnus expansion for the solution of a homogeneous linear ODE.

In this work, we consider a linear age-structured problem with diffusion and non-homogeneous boundary conditions both for the age space variables. We handle by re-writing it as non-densely defined abstract Cauchy problem. To that aim develop new result on closedness of commutative sum two operators using theory integrated semigroups. As an application result, are able to associate suitable semi...