On Mixed Problems for Parabolic Equations in General Holder Spaces
MARTÍN LOPEZ MORALES
Departamento de Computación
Tecnológico de Monterrey. Campus Ciudad de México
Calle del Puente 222. Ejidos de Huipulco. Tlalpan. 14380.México D.F.
MEXICO
Abstract: - We consider the second and the third boundary value problems for linear parabolic equations in a cylindrical domain .We establish new a priori estimates in general Hölder norms for the solutions to these problems, under the assumption that the coefficients satisfy the general Holder condition with respect to the space variables only. In this connection, however, we also obtain an estimate of the modulus of continuity with respect to the time of the second derivatives with respect to x of the corresponding solutions.
On the basis of new a priori estimates for the solutions to these problems, we establish the corresponding solvability theorems in general Hölder spaces.
We apply these results to obtain the solvability to these problems for nonlinear parabolic equations
Key-Words: - estimates, solvability, equations, parabolicity, solution, problem
8
1. Introduction
We consider the linear parabolic equation
in a cylindrical domain with the initial condition
(2)
and one of the following conditions on the lateral surface of
(3)
(3´)
Here is a point of is a bounded domain in n-dimensional Euclidean space
where is the boundary of
.
We establish new a priori estimates in general anisotropic Hölder norms for the solutions of the problems, (1), (2), (3) and (1), (2), (3´) under the assumption of the general anisotropic Holder continuity of the coefficients with respect to space variables only. In this connection, however, we also obtain an estimate of the modulus of continuity with respect to the time of the leading derivatives (but not ).
Note that in the works [ 1 ] - [ 11] and in many others the a priori estimates are obtained under the fulfilment Hölder condition with respect to the totality of variables () on the coefficients of Equation (1).On the basis of new a priori estimates for the solutions of the problems (1), (2), (3) and (1), (2), (3´) we establish the corresponding theorems on the solvability for these problems in general anisotropic Hölder spaces.
We assume that the coefficients of Equation (1) satisfy the uniform parabolicity condition: for any nonzero vector and (4)
We apply the results in the linear theory to establish the local solvability with respect to the time t, in general Hölder anisotropic spaces, to the boundary valued problem for the nonlinear parabolic equation
(5)
in with the initial condition (2) and one of the boundary conditions (3) or (3´), where
In the present work, like in the works the equation (5) is linearized directly. No conditions are imposed here on the nature of the growth of the non linearity of the function A , which is defined for The main assumption concerning to the function ,where and is the parabolicity condition:for any non zero vector and any (6)
where here and below denote the usual scalar product in
In all the work we suppose that in the equation (1), the functions and satisfy the general Hölder condition in of exponent with respect to the space variables only and satisfies the general Hölder condition in of exponent with respect to the space variables only.
All the coefficients and the independent terms are continuous in the cylinder .
In the case of the problems (5) , (2) , (3) and (5) , (3) , (3`) we require less smoothness conditions from the functions
than in the works [ 8 ] , [ 9 ] and [ 15 ].
Some close results have been established in [14] [16] and [21]-[23].
The results of this work must find applications in the problems of heat conduction, diffusion, Mechanics of Fluids and many others.
.2.Notations and Propositions
We shall say that the function defined in the cylinder satisfies the general Hölder condition in of exponent with respect to the space variables, if there exists a constant
C > 0 such that
.
The function is defined and continuous in . Moreover it has the following properties:
Ia.
Ib.
Ic. If then and where is sufficiently small number (we suppose that the derivative exists and it is a continuous function in .
Note that the condition Ic introduces a new set of functions ( the functions that satisfy the general Hölder condition with respect to space variables only ).In this new set of functions we will obtain the corresponding existence and uniqueness theorems for the solutions to the problems (1),(2),(3); and (5) , (2) , (3) and (5) , (3) ,(3’)
We denote by 1,2, the set of functions for which
For the functions we introduce the functions
The functions are functions of the type See examples of functions of the type in [12].
For the functions defined and Holder continuous ( in the general sense ) in the cylinder of exponent , with respect to space variables, we introduce the following norms :
(7)
where
(8)
. (9)
For the functions that have continuous derivatives with respect to x up to the order inclusively in the cylinder and satisfy the general Holder condition of exponent , with respect to the space variables in the cylinder , we define the norms
(10)
(11)
,m=0,1,2. (12)
We will denote by the Banach space of functions that are continuous in , together with all derivatives respect to x up to the order inclusively and have a finite norm (12).
We define the parabolic distance between each two points by the magnitude
(13)
For the functions that have continuous derivatives with respect to x up to the order inclusively in the cylinder we consider the usual norms
(14)
(15)
(16)
(17)
We denote by the Banach space of functions that are continuous in together with all derivatives respect to up to the order and have finite norms (15) - (17), respectively.
It is possible to consider all the preceding definitions in the layer (see ) or in any domain contained in .
With respect to the coefficients of the equation (1) we assume that
and
(18)
moreover the coefficients are uniformly continuous with respect to t on for any there exists such that for all
we have
(19)
Definition We will say that if for every point there exists an n-dimensional ball B with centre such that can be represented for some in the form
and the function has second derivatives that are bounded and Hölder continuous of exponent .
In what follows, we assume that , then there exists (by the Heine-Borel lemma) a number , such that the part of the boundary lying in the ball
is represented by one of the equations
where the point belongs to the domain
For the functions
we introduce the following norms
(20)
where the norms in the right- hand side is considered in the cylinder .
If , we will say that
Moreover we will consider in the norm
(21)
With respect to the boundary " data " in the problems ( 1 ) , ( 2 ) , ( 3 ) and (1), ( 2) , (3´) we assume that
,
and
(22)
For equation ( 5 ) we consider in addition to the parabolicity condition ( 6 ) that there exists a domain
in which the function together with its derivatives with respect to , up to the second order inclusively is continuous and satisfies the Lipschitz condition with respect to and a general Hölder condition of exponent with respect to and with the constant .Moreover
All the mentioned derivatives are bounded in by the constant .
On the lateral surface the function are uniformly continuous with respect to t
(see [18] ).
Now we shall consider the equation (1) with the initial zero condition
(23)
and one of the boundary conditions (3) or ( 3' ).
3. Bounds for solutions to the mixed problems
Theorem 1.Let
be a solution of the problem (1), (23), (3) in the cylinder . Assume that
if or ,
and the conditions (4) , (18) , (19) hold . Then the following inequality holds for .
(24)
K denotes here and below a constant depending on
.
Proof. Let be an arbitrary point of since then there exists two numbers such that the part of the boundary lying in the ball is representable by one of the equations
Denote by the cylinder The transformations
(25)
maps onto a set which lies on and, as we may assume, it maps onto a domain lying in the half space and some half ball is contained in this domain. The mapping (25) is one-to-one, non degenerated and its transforms the equation (1) into the equation of the same type.
With the aid of the results to the Cauchy problem ( see [ 12 ] ), the properties of the function , the estimates for the solutions to the mixed problems for parabolic equations in the a half space ( see [8] and [9] the interpolation inequalities of Lemma 2 in [12] with e small enough we get the estimates
(26) (27)
where
This means that the estimate ( 24 ) holds for . Similarly the estimate (24) yields for then by means of a finite number of steps with length we run over the interval . This complete the proof of theorem 1
.
Remark 1. By virtue of Lemma 4 in the work [16] ( estimates for the solution to the Cauchy problem parabolic equations ) , with the aid of theorem 1 in [16] and using the interpolation inequalities of Lemma 2 in [12 ] we conclude that for
(28)
Remark 2. We can reduce the mixed problem ( 1 ) , ( 2 ) , ( 3 ) to the mixed problem ( 1 ),
( 23 ) , ( 3 ) by means of the transformation
Theorem 2 Let be the solution of the problem (1 ) , ( 23 ) , ( 3` ) in the cylinder . Assume that if or ,, and conditions (4) , (18) and (19) hold . Then the following inequality holds for
(29)
K is a constant depending on The Prof. of theorem 2 is similar to the proof of theorem 1.
Remark 3. Arguing as in the proof of Remark 2 we can get the estimate
(30)
Remark 4. We can reduce the mixed problem ( 1 ) , ( 2 ) , ( 3´ ) to the mixed problem ( 1 ),
( 23 ) , (3´ ) by means of the transformation
4. Existence and uniqueness theorems
Theorem 3 . Suppose that all conditions of the Theorem 1 are true. Assume, furthermore, that the following consistency condition holds
(31)
Then there exists a unique solution to the problem (1), (23), (3) in the space with a continuous derivative .
Theorem.4 Suppose that all assumptions of the Theorem 2 hold. Moreover, the next consistency condition is satisfied
(32)
Then there exists a unique solution to the problem (1), (2), (3') in with a continuous derivative .
We can get the proof to these theorems on the basis of our new a priori estimates established in this work and with the aid of the method of continuity in a parameter (see [5] and [20] ).
We proceed now to formulate the local existence theorem for solutions to the non-linear problems for the equation (5).Here we consider that the function
With respect to the initial and boundary functions in the problems (5), (2), (3) and (5), (2), (3') we assume that
(33)
and
(34)
Theorem 5 Suppose that all assumptions with respect to the function hold and the following consistency condition is satisfied
(35)
Then there exists , to determined by the above assumptions such that the problem (5), (23),(3) has in the cylinder with a unique solution with a continuous derivative
The proof of the theorem 5 is similar to the proof of the theorem 3, in the work [14 ] and to the Prof. of theorem 3 in the work [18].
Remark.5 We can reduce the mixed problem (5), (2), (3) to the problem with zero initial condition (5), (23), (3), by the means of the transformation
Theorem 6 Suppose that all assumptions with respect to the function hold and the following consistency condition is satisfied
Then there exists determined by the above assumptions such that the problem (5), (26), (3') has in the cylinder a unique solution with a continuous derivative
The proof of this theorem is similar to the proof of the theorem 5.
5. REFERENCES
[1] R. B. Barrar, "Some estimates for
solutions of parabolic equations", J. Math. Anal. App. (1961) 373 - 397.
[2] C. Ciliberto, "Formule di maggiorazine e teoremi di esistenza per le soluzione della equazioni paraboliche in due variabili", Ricerche Mat. 3 (1945) 40 - 75.
[3] L.I "Kaminin , V. N. Masliennikova, "Boundary estimates of solution of third boundary value problem for a parabolic