Cauchy Problem For Evolution Equations Research Articles

Cauchy type problems for fractional differential equations

While it is known that one can consider the Cauchy problem for evolution equations with Caputo derivatives, the situation for the initial value problems for the Riemann–Liouville derivatives is less understood. In this paper, we propose new type initial, inner, and inner-boundary value problems for fractional differential equations with the Riemann–Liouville derivatives. The results on the existence and uniqueness are proved, and conditions on the solvability are found. The well-posedness of the new type of initial, inner, and inner-boundary conditions is also discussed. Moreover, we give explicit formulas for the solutions. As an application fractional partial differential equations for general positive operators are studied.

Radon measures as solutions of the Cauchy problem for evolution equations

We consider a standard Navier–Stokes system on the n-D torus $${\mathbb {T}}^n={\mathbb {R}}^n/{\mathbb {Z}}^n$$ , valid when the flow is not very compressible and the temperature does not vary too much. We construct a sequence of approximate solutions that tend to satisfy the equations in a weak sense for arbitrary physical initial conditions. By weak compactness, we obtain Radon measure solutions in density and momentum when the velocity of the flow is finite, as numerically observed in all tests, and the absence of void regions in the flow. We notice that the method also applies without viscosity and complements results on other systems of evolution equations such as the systems of isothermal and isentropic gas flows considered in Colombeau (Z Angew Math Phys 66(5):2575–2599, 2015).

Asymptotic behavior of $$C^0$$ C 0 -solutions of evolution equations with uncertainties

In this paper, we study Cauchy problem for evolution equations with uncertainties in semilinear metric space $$C(J_\infty ,\mathcal {T}_{\mathcal {F}})$$ of symmetric triangular fuzzy-valued functions. We use fixed point approach and measure of noncompactness to investigate the solvability, the existence of decay $$C^0$$ -solutions and the asymptotic stability of equilibrium point of the problem. We present several examples to illustrate the theoretical results.

Probabilistic Representation of a Solution of the Cauchy Problem for Evolution Equations with Riemann--Liouville Operators

This paper studies properties of probabilistic approximation of a solution of the Cauchy problem for evolution equations with fractional differential operators of order more than two. To this end we construct analogous one-sided $\alpha$-stable distributions for noninteger $\alpha>2$. Although densities of these distributions are signed functions, using generalized functions methods, it is possible to give them an exact probability sense.

Some properties of solutions of the cauchy problem for evolution equations

We consider the Cauchy problem for a first-order operator-differential equation with singular data. The results are used to study boundary value problems for parabolic equations with operator-valued coefficients.

The cauchy problem for evolution equations with the Bessel operator of infinite order. II

The cauchy problem for evolution equations with the Bessel operator of infinite order. II

The Cauchy problem for evolution equations with the Bessel operator of infinite order. I

We establish necessary and sufficient conditions under which the Bessel operator of infinite order is bounded in certain spaces. We study properties of the Bessel transformations of distributions from these spaces, those of convolutions, convolutors, and multiplicators.

The Cauchy problem for 𝑝-evolution equations

In this paper we deal with the Cauchy problem for evolution equations with real characteristics. We show that the problem is well-posed in Sobolev spaces assuming a suitable decay of the coefficients as the space variable x → ∞ x\to \infty . In some cases, such a decay may also compensate a lack of regularity with respect to the time variable t t .

Energy methods for Cauchy problems of evolutions equations

In the present paper a numerical method based on minimizing energy functionals, is developed for solving Cauchy problem of evolution equations. Two cases are treated: the parabolic equations as the heat equations and the hyperbolic ones as the elastodynamics equations. The method is first presented in some details, then, illustrated on various applications.

Cauchy problem for evolution equations with an infinite-order differential operator: II

Cauchy problem for evolution equations with an infinite-order differential operator: II

Properties of solutions of the Cauchy problem for essentially infinite-dimensional evolution equations

We investigate properties of solutions of the Cauchy problem for evolution equations with essentially infinite-dimensional elliptic operators.

Essentially Infinite-Dimensional Evolution Equations

We investigate the Cauchy problem for evolution equations with essentially infinite-dimensional elliptic operators.

Optimal well-posedness of the Cauchy problem for evolution equations with $C^N$ coefficients

We deal with the Cauchy problem for a $2$-evolution operator of Schrödinger type with $C^N$ coefficients in the time variable, $N>2$. We find the Levi conditions for well-posedness in Gevrey classes of index $1/2 + N/4$, which is the best possible, as we show by means of counterexamples.

Cauchy Problem for Evolution Equations of Schrödinger Type

In (R. Agliardi, 1995, Internat. J. Math.6, 791–804) we proved the well-posedness of the Cauchy problem in H∞ for some p-evolution equations (p⩾1) with real characteristic roots. For this purpose some assumptions on the lower order terms are needed, which, in the special case p=1, recapture well-known results for hyperbolic operators. In (R. Agliardi, 1995, Internat. J. Math.6, 791–804) the leading coefficients are assumed to be constant. In this paper we allow them to be variable. Our result is applicable to 2-evolution differential operators with real characteristics, i.e., to Schrödinger type operators. This class of operators comprehends, for example, Schrödinger operator Dt−Δx or the plate operator D2t−Δ2x. The Cauchy problem in H∞ for such evolution operators has been studied extensively by Takeuchi when the coefficients in the principal part are constant and the characteristic roots are distinct.

On a necessary condition for the wellposedness of the Cauchy problem for evolution equations

On a necessary condition for the wellposedness of the Cauchy problem for evolution equations

A finite-difference method of solving the Cauchy problem for evolution equations

The correctness and convergence of the finite-difference method of solving the Cauchy problem for a second-order equation in Hilbert space is investigated. The wave equation with mixed derivative is presented as an application.

Uniqueness and Norm Convexity in the Cauchy Problem for Evolution Equations with Convolution Operators

Uniqueness and Norm Convexity in the Cauchy Problem for Evolution Equations with Convolution Operators

Generalizations of Results of Agmon and Nirenberg to the Cauchy Problem for Evolution Equations

Uniqueness and norm convexity for evolution equations are investigated by using tools developed by Agmon and Nirenberg.

Uniqueness and norm convexity in the Cauchy problem for evolution equations with convolution operators

Uniqueness in the Cauchy problem is shown under suitable conditions for evolution equations of the form u t ( x , t ) − B ( t , D x ) u ( x , t ) = 0 {u_t}(x,t) - B(t,{D_x})u(x,t) = 0 , where B is a pseudo-differential operator of order k ≧ 0 k \geqq 0 in the x variables. This is proved as a corollary to a norm convexity relation. In the process of showing this, an extension to Hölder’s inequality is derived.

Generalizations of results of Agmon and Nirenberg to the Cauchy problem for evolution equations

Uniqueness and norm convexity for evolution equations are investigated by using tools developed by Agmon and Nirenberg.