Research Article Open Access

The Regularity of the Solutions to the Cauchy Problem for the Quasilinear Second-Order Parabolic Partial Differential Equations

Mykola Yaremenko1
  • 1 Partial Differential Equation, Ukraine
Journal of Mathematics and Statistics
Volume 16 No. 1, 2020, 76-89

DOI: https://doi.org/10.3844/jmssp.2020.76.89

Submitted On: 1 April 2020
Published On: 2 June 2020

How to Cite: Yaremenko, M. (2020). The Regularity of the Solutions to the Cauchy Problem for the Quasilinear Second-Order Parabolic Partial Differential Equations. Journal of Mathematics and Statistics, 16(1), 76-89. https://doi.org/10.3844/jmssp.2020.76.89

Abstract

This article is dedicated to expanding our comprehension of the regularity of the solutions to the Cauchy problem for the quasilinear second-order parabolic partial differential equations under fair general conditions on the nonlinear perturbations. In this paper have been obtained that the sequence of the weak solutions uz ∈ V1,02, z = 1,2,..... to the Cauchy problems for the Equations (15) under the initial conditions uz (0,x) = φ0z converges to the weak solution to the Cauchy problem for the Equation (1) under the initial condition u(0, x) = u0 in V1,02.

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Keywords

  • Quasi-Linear Partial Differential Equations
  • Nonlinear Partial Differential Equations
  • Parabolic
  • Nonlinear Operator
  • Weak Solution
  • A Priori Estimations