On the branching of solutions and Signorini's perturbation procedure in elasticity
On the Caginalp system with dynamic boundary conditions and singular potentials
This article is devoted to the study of the Caginalp phase field system with dynamic boundary conditions and singular potentials. We first show that, for initial data in , the solutions are strictly separated from the singularities of the potential. This turns out to be our main argument in the proof of the existence and uniqueness of solutions. We then prove the existence of global attractors. In the last part of the article, we adapt well-known results concerning the Łojasiewicz inequality in...
On the Calabi-Yau equation in the Kodaira-Thurston manifold
We review some previous results about the Calabi-Yau equation on the Kodaira-Thurston manifold equipped with an invariant almost-Kähler structure and assuming the volume form T2-invariant. In particular, we observe that under some restrictions the problem is reduced to aMonge-Ampère equation by using the ansatz ˜~ω = Ω− dJdu + da, where u is a T2-invariant function and a is a 1-form depending on u. Furthermore, we extend our analysis to non-invariant almost-complex structures by considering some...
On the Cauchy problem and initial traces for a class of evolution equations with strongly nonlinear sources
On the Cauchy problem for a class of parabolic equations with variable density
The well-posedness of the Cauchy problem for a class of parabolic equations with variable density is investigated. Necessary and sufficient conditions for existence and uniqueness in the class of bounded solutions are proved. If these conditions fail, sufficient conditions are given to ensure well-posedness in the class of bounded solutions which satisfy suitable constraints at infinity.
On the Cauchy problem for a degenerate parabolic differential equation.
On the Cauchy problem for a second order semilinear parabolic equation with factored linear part.
On the Cauchy problem for convolution equations
We consider one-parameter (C₀)-semigroups of operators in the space with infinitesimal generator of the form where G is an -valued rapidly decreasing distribution on ℝⁿ. It is proved that the Petrovskiĭ condition for forward evolution ensures not only the existence and uniqueness of the above semigroup but also its nice behaviour after restriction to whichever of the function spaces , , p ∈ [1,∞], , a ∈ ]0,∞[, or the spaces , q ∈ ]1,∞], of bounded distributions.
On the Cauchy problem for hyperbolic functional-differential equations
We consider the Cauchy problem for a nonlocal wave equation in one dimension. We study the existence of solutions by means of bicharacteristics. The existence and uniqueness is obtained in topology. The existence theorem is proved in a subset generated by certain continuity conditions for the derivatives.
On the Cauchy problem for linear hyperbolic functional-differential equations
We study the question of the existence, uniqueness, and continuous dependence on parameters of the Carathéodory solutions to the Cauchy problem for linear partial functional-differential equations of hyperbolic type. A theorem on the Fredholm alternative is also proved. The results obtained are new even in the case of equations without argument deviations, because we do not suppose absolute continuity of the function the Cauchy problem is prescribed on, which is rather usual assumption in the existing...
On the Cauchy problem for linear partial differential-functional equations of first order
On the Cauchy problem for linear PDEs with retarded arguments at derivatives
We present an existence theorem for the Cauchy problem related to linear partial differential-functional equations of an arbitrary order. The equations considered include the cases of retarded and deviated arguments at the derivatives of the unknown function. In the proof we use Tonelli's constructive method. We also give uniqueness criteria valid in a wide class of admissible functions. We present a set of examples to illustrate the theory.
On The Cauchy Problem for Non Effectively Hyperbolic Operators, The Ivrii-Petkov-Hörmander Condition and the Gevrey Well Posedness
2000 Mathematics Subject Classification: 35L15, Secondary 35L30.In this paper we prove that for non effectively hyperbolic operators with smooth double characteristics with the Hamilton map exhibiting a Jordan block of size 4 on the double characteristic manifold the Cauchy problem is well posed in the Gevrey 6 class if the strict Ivrii-Petkov-Hörmander condition is satisfied.
On the Cauchy problem for the equation of one-dimensional non-stationary filtration with non-continuous initial data
On the Cauchy problem for the equations of ideal compressible MHD fluids with radiation
We consider a system of balance laws describing the motion of an ionized compressible fluid interacting with magnetic fields and radiation effects. The local-in-time existence of a unique smooth solution for the Cauchy problem is proven. The proof follows from the method of successive approximations.
On the Cauchy Problem for the Kadomstev-Petviashvili Equation.
On the Cauchy problem for the Yang-Mills field equations
On the Cauchy Problem in Nonlinear 3-d-Thermoelasticity.
On the Cauchy problem of a quasilinear degenerate parabolic equation.
On the Cauchy-problem for generalized Kadomtsev-Petviashvili-II equations.