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

Email: math.kiev@gmail.com 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 u  V1,0, z = 1,2,..... to the Cauchy problems for the Equations (15) under the initial conditions u (0,x) = 0 converges to the weak solution to the Cauchy problem for the Equation (1) under the initial condition u(0, x) = u0 in V1,0.


Introduction
Let us consider the quasilinear second-order parabolic partial differential equations:   , ,.., ( , , ) , , , , under the initiation conditions: where the u(t,x) is the unknown function,  > 0 is a real number and f(t,x) = f is a given function. The term b (t, x, u, u) is a measurable function of four arguments.
The matrix aij (t,x,u) is a measurable elliptical matrix l  l size such that there is a number :     (2) for almost every [ , ] tT  0 and l xR  . Or we will consider a more restrictive condition: ,..., ( , , ) (1)  holds for almost every [ , ] tT  0 , l xR  and for all , q vW  10 . The main object of this paper is the regularity properties of the solutions to the quasilinear parabolical partial differential Equation (1) under the conditions that its coefficients belong to the certain functional classes and functional spaces.

Definition A real-valued function u(t,x) is called a weak solution to the parabolical partial differential Equation
The conditions of linear growth: 1. b (t, x, y, z) is a measurable function of its arguments and   l loc b L R  1 2. Function b (t, x, y, z) t [0, T] satisfies inequality: and >0 is a form-boundary and c() The general information on the partial differential equations and the existence of their solutions can be found in the extensive literature on the conditions on their coefficients under which there are the solutions of these equations in a specific functional space (Adams and Hedberg, 1996;Gilbarg and Trudinger, 1983;Ladyzenskaja et al., 1968;Nirenberg, 1994;Veron, 1996;Yaremenko, 2017a;2017b). O. Ladyzhenskaya, N. Uraltseva, O.A. Solonnikov developed the Ennio de Giorgi's method (DeGiorgi, 1968) for establishing a priory estimation of the solution of such equations. 1960 J. Moser enhance the maximum principle and created a new method of studying the regularity of the solutions of elliptic differential equations and Harnack's inequality under the assumption that the coefficients are bounded measurable and satisfy a uniform ellipticity condition, these results were summarized in the work of Ladyzenskaja et al. (1968).
A Lebesgue space L p (R l , d l x) for 1< p <  can be defined as a set of all real-valued measurable functions defined almost everywhere such that the Lebesgue integral of its absolute value raised to the p-th power is a finite number with its natural norm: We will use the inequality:  0 and its consequence: The f  L p yields f | f | p-2 L q that justify the last equation (Gilbarg and Trudinger, 1983;Ladyzenskaja et al., 1968

The Estimation of the Solutions to the Equation (1)
For almost every t  [0, T], let us consider the integral identity: and estimate: From (1) under the conditions (4) we obtain (6). Next, we estimate every term separately: , using Young and Holder inequalities are obtaining: For almost all t applying  In case of p = 2 there is the next estimation: Let u(t,x) be a weak solution. We denote ( , ) h v t x the average of function v(t,x) at t by formulae: we transform: where the function v(t,x) is tautological equals zero over

Remark
The order of averaging and differentiation by x are interchangeable.
and we have:

A Priori Estimation of the Solution to (1)
Let us assume that ellipticity condition and (4) ,, , and integrate by parts, we are obtaining where the K() is a cube in R l with an edge length of .