Existence and regularity of solution of the heat equation in periodic Sobolev spaces
Keywords:Semigroups theory, Heat equation, nonhomogeneous equation, Periodic Sobolev spaces, Fourier theory
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  and Santiago and Rojas . 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.
Iorio R. and Iorio V. Fourier Analysis and partial differential equation. Cambridge University, 2001.
Santiago Y. Rojas S., Quispe T. Espacios de Sobolev periódico y un problema de Cauchy asociado a un modelo de ondas en un fluido viscoso, Theorema, Segunda Época 3(4) (2016) 7-23.
Santiago, Y. Rojas, S. Regularity and wellposedness of a problem to one parameter and its behavior at the limit, Bulletin of the Allahabad Mathematical Society 32(2) (2017) 207-230.
Rubinstein, I. and Rubinstein, L. Partial differential equations in classical mathematical physics. Cambridge University Press, 1998.
Shahjalal M., Sultana, A., Valluri R., Mitra, N. and Khan, A. Black-Scholes PDE and Ornstein-Uhlenbeck SDE process to analyse stock option: A study in Fuzzy context, Intern. Journal of Mathematics and Computing 1(1) (2015) 1-10.
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