Enunciamos y damos una prueba de la desigualdad de Poincaré cuando el dominio está acotado en una dirección, usando el caso unidimensional donde hacemos uso del cálculo elemental. Damos consecuencias importantes en los espacios de Sobolev, y hacemos un estudio del dominio donde vale o no la desigualdad de Poincaré, usando el Lema de Equivalencia.

In this article we prove that the Cauchy problem associated to the heat equation in periodic Sobolev spaces is well posed. We do this in an intuitive way using Fourier theory and in a fine version using Semigroups theory, inspired by works Iorio [1] and Santiago and Rojas [3]. Also, we study the relationship between the initial data and differentiability of the solution.Finally, we study the corresponding nonhomogeneous problem and prove it is locally well posed and even more we obtain the continuous dependence of the solution with respect to the initial data and the non homogeneity.

In this article, we first prove that the initial value problem associated to the homogeneous wave equation in periodic Sobolev spaces has a global solution and the solution has continuous dependence with respect to the initial data, in [0; T], T > 0. We do this in an intuitive way using Fourier theory and in a fine version introducing families of strongly continuous operators, inspired by the works of Iorio [4] and Santiago and Rojas [7].Also, we prove that the energy associated to the wave equation is conservative in intervals [0; T], T > 0.As a final result, we prove that the initial value problem associated to the non homogeneous wave equation has a local solution and the solution has continuous dependence with respect to the initial data and the non homogeneous part of the problem.

In this article, we prove the uniqueness solution of the homogeneous and non-homogeneous heat equation in periodic Sobolev spaces. We do it in a different way from what we did in [3], in this case we perform differential calculus in Hs-per and we take advantage of the immersion and properties of periodic Sobolev spaces. With this proof we gain to visualize the dissipative property of the homogeneous problem and with this we deduce the continuous dependence with respect to the initial data and the uniqueness solution for both cases: homogeneous and non-homogeneous.

Enunciamos y damos una prueba de la desigualdad de Poincaré cuando el dominio está acotado en una dirección, usando el caso unidimensional donde hacemos uso del cálculo elemental. Damos consecuencias importantes en los espacios de Sobolev, y hacemos un estudio del dominio donde vale o no la desigualdad de Poincaré, usando el Lema de Equivalencia.

In this article we prove that the Cauchy problem associated to the heat equation in periodic Sobolev spaces is well posed. We do this in an intuitive way using Fourier theory and in a fine version using Semigroups theory, inspired by works Iorio [1] and Santiago and Rojas [3]. Also, we study the relationship between the initial data and differentiability of the solution.Finally, we study the corresponding nonhomogeneous problem and prove it is locally well posed and even more we obtain the continuous dependence of the solution with respect to the initial data and the non homogeneity.

In this article, we first prove that the initial value problem associated to the homogeneous wave equation in periodic Sobolev spaces has a global solution and the solution has continuous dependence with respect to the initial data, in [0; T], T > 0. We do this in an intuitive way using Fourier theory and in a fine version introducing families of strongly continuous operators, inspired by the works of Iorio [4] and Santiago and Rojas [7].Also, we prove that the energy associated to the wave equation is conservative in intervals [0; T], T > 0.As a final result, we prove that the initial value problem associated to the non homogeneous wave equation has a local solution and the solution has continuous dependence with respect to the initial data and the non homogeneous part of the problem.

In this article, we prove the uniqueness solution of the homogeneous and non-homogeneous heat equation in periodic Sobolev spaces. We do it in a different way from what we did in [3], in this case we perform differential calculus in Hs-per and we take advantage of the immersion and properties of periodic Sobolev spaces. With this proof we gain to visualize the dissipative property of the homogeneous problem and with this we deduce the continuous dependence with respect to the initial data and the uniqueness solution for both cases: homogeneous and non-homogeneous.

In this articlewe prove that the Cauchy problemassociated to the Schrödinger equation in periodic Sobolev spaces is well posed. We do this in an intuitiveway using Fourier theory and in a fine version using Groups theory, inspired by works Iorio [3], Santiago and Rojas [12] and [13]. Also, we study the relationship between initial data and differentiability of the solution. Finally, we study the corresponding non-homogeneous problemand prove that it is locallywell posed, and that the solution has continuous dependence with respect to the initial data and the non-homogeneity in  compact intervals.

In this work, we study the Fourier Theory in the space of periodic distributions: P’. We analyze the existence of at least one solution for the distributional differential problem in connection with the zeros of a polynomial. We prove that there are infinite solutions when the Fourier coefficients vanish at the integer zeros of the polynomial and otherwise does not have solution. We deduce the existence and uniqueness by considering that the polynomial lacks integer zeros. In the cases of existence, we deduce the analytical solutions. Moreover, we get a result firelated with the continuous dependence of the solution. Finally, we give some conclusions and applications.

In this article, we prove the existence and uniqueness of the solution of the homogeneous  generalized Schrödinger equation of order m in the periodic distributional space P0, where m is an even number not a multiple of four. Furthermore, we prove that the solution depends continuously respect to the initial data in P0. Introducing a family of weakly continuous operators, we prove that this family is a group in P0. Then, with this family of operators, we get a fine version of the existence and dependency continuous theorem obtained. Finally, we give the conclusions and remarks derived from this study.

In this work we begin by studying the generalized multiplication operator M on the l2(Z). We prove that this operator is not bounded, is densely defined and symmetric and therefore does not admit a symmetric linear extension to the entire space. We introduce a family of operators on the l2(Z) space with n even and demonstrate that it forms a contraction semigroup of class Co, having −M as its infinitesimal generator. We also prove that if we restrict the domains of that family of operators, they still remain a contraction semigroup. Finally, we give results of existence of solution of the associated abstract Cauchy problem and properties of continuous dependence of the solution in connection to other norms.

We prove the existence of solution for a mathematic model of diffusion of a contaminant using the Nonlinear Semigroups Theory, by means of afin operators. We also study a realistic model by means of satura tíon effects.

In this paper we show the global existence and the exponencial decay of solutions of the one-dimensional wave equation with localized frictional damping.

We proved that the semigroup C0 associated to a viscoelastic system is analytic and exponentially stable.

We prove that the eo semigroup, associated to athermoelastic system is analitic and exponential stable

In this article, we prove the existence and uniqueness of the weak solution of a nonlinear wave equation. We prove the uniqueness by using the Visik - Ladyshenkaia Method. Also, using the Nakao s Lemma, we prove the exponential decay ofthe energy associated to the system.

In this work, we consider the problem of existence of global solutions for a scalar wave equation with dissipation. We study also, the asymptotic behaviour of the solutions. In this part of the paper; we present an alternative method - inspired in nonlinear techniques. In order lo attack this problem, specifically, we used the Conrad - Rao method [l].

En este trabajo se enuncia y demuestran dos Lemas importantes de Nakao. Estos Lemas nos permitirán obtener la tasa de decaimiento de la solución de algunas ecuaciones en derivadas parciales. En particular, probamos el decaimiento exponencial de la solución global de una ecuación de onda.

Using semigroup theory we are going to prooue the global existence solution o] a evolution model, which represente the magneto-elastic waves in a boundary conduciive media. Also, we are going to prooue that the the system is well set and using Dinamic Systems and La Salle 's Invariance Principie we are going to show that the energy asociated to the system decays to zero when t→+ ∞.