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ISSN: 2168-9679
Journal of Applied & Computational Mathematics
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Regularity of Solutions of Degenerate Parabolic Non-linear Equations and Removability of Solutions

Gadjiev TS*, Yangaliyeva A and Zulfalieva G

Institute of Mathematics and Mechanics of NAS of Azerbaijan, AZ1141 Baku, Azerbaijan

*Corresponding Author:
Gadjiev TS
Institute of Mathematics and
Mechanics of NAS of Azerbaijan
AZ1141 Baku, Azerbaijan
Tel:
+994506728756
E-mail:
[email protected]

Received Date: July 07, 2017; Accepted Date: August 21, 2017; Published Date: August 31, 2017

Citation: Gadjiev TS, Yangaliyeva A, Zulfalieva G (2017) Regularity of Solutions of Degenerate Parabolic Non-linear Equations and Removability of Solutions. J Appl Computat Math 6: 364. doi: 10.4172/2168-9679.1000364

Copyright: © 2017 Gadjiev TS, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

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Abstract

In this paper we prove regularity of solutions of degenerate parabolic nonlinear equations. We also the proof of a removability theorem for solutions to degenerate parabolic nonlinear equations.

Keywords

Degenerate; Nonlinear parabolic equations; Regularity; Removability

Introduction

Let we are considered in cylindrical domains QT=Ω × (0,T), where Ω ⊂ Rn, n ≥ 2 is a bounded Lipschitz domain, T>0, degenerate nonlinear parabolic equations

ut−div(ω(x)|Du|p−2Du)=0 (1.1)

u|Γ(QT)=h, (1.2)

where Γ(QT)=(Ω¯ × {0}) ∪ (∂Ω × [0,T]) denote the parabolic boundary of QT, h : QT→R continuous function, ω(x)-Makenxhoupt weight function [1].

To regularity of solutions to the degenerate parabolic non-linear operator introduced by DiBenedetto et al. [2,3]. Let Equation weighted space, where norm following:

Equation

where the parabolic metric is defined as

Equation

Main Results

We are now ready to state our result which concerns regularity for solutions to the problem (1.1), (1.2).

Theorem 2.1

Let’s consider problem (1.1), (1.2)and let u(x,t) solve this problem. Let Q0T ⊂ QT be a bounded space time cylinder such that (interior regularity)

Equation (2.1)

Theorem 2.1 concerns optimal interior regularity. We also establish optimal regularity up to initial state. In particular, in this case we prove Equation estimates on Equation for every Ω, ⊂ Ω. We doing remark that in this case Equation is not a compact subset of QT .

In this context hold following result [1-12].

Theorem 2.2

Let u(x,t) solve problem (1.1), (1.2) and(Initial time regularity)

Equation (2.2)

We also can be is considered obstacle problem similarly to problem (1.1), (1.2). In the case of linear uniformly parabolic equations [4]. Optimal regularity problem of the solution is considered [5].

We are study weak solutions from Equation space. In the space

Equation the norm denote the space of equivalence classes of functions f with distributional gradient Df, both of which are pth power integral on QT . Let

Equation

be the norm in Equation

Given t1<t2 we denote by Equation the space of functions such that for almost every t, t1 ≤ t ≤ t2 the function

Equation

We say that a function u(x, t) is a weak solution to (1.1), (1.2) in an open set

QT ⊂ Rn+1 if whenever Q0T0 × (t1,t2) ⊂ QT with Ω0 ⊂ Ω ⊂ Rn and t1<t2 then Equation and

Equation (2.3)

for all nonnegative Equation.

Using Theorem2.1 we are able to establish sharp removability conditions for compact sets. We of cylinders introduced

Equation

And a concave modulus of continuity ψ(·). We let ψ: R+ → R+ be a concave modulus of continuity, i.e., concave non-decreasing function such that ψ(1)=1 and Equation. We also define Hausdorff measure as follows. We let for fixed δ,0< δ<r0 and Equation be a family of cylinders such that Equation and 0 < ri < δ for i=1,2,..

Using this notation we let

Equation

where the indium is taken with respect to all possible coverings L(δ,ψ(·); E) of E.

Theorem 2.3

Let QT be a cylindrical domain and let E ⊂ QT be a closed set. Let u(x,t) is a weak solution to eqn. (1.1) in QT \E and that Equation

Assume also that Hψ(·)(E)=0. Then the set E is removable, i.e., u(x,t) can be extended to be a weak solution in QT .

Similarly result the fundamental work [6], under assumption Holder continuity of the solution can be found [7-12].

Proof of theorem 2.1

We assume Q,T ⊂ QT such that Q0T ∩ Γ(QT )=∅.

We define function

Equation

Then Equation. Let ¯u be the unique solution to

Equation

u¯(x,t)=h¯(x,t) on Γ(Ω × (0,∞)).

By the uniqueness ¯u=u in Ω × [0,T] and hence ¯u is an extension of u. Let

R=max{1,diamΩ,T1/2}. As clearly

T ≤ (ψ(R))2−p RpR2.

Whenever R ≥ 1. By maximum and minimum principle implies that

Equation (2.4)

We may assume that Equation, where Ω0 ⊂ Ω and τ>0. We let R be

a number subject to the restrictions

Rdist0,∂Ω ),τRp max{osch,ψ(R),s · R}2−p.

QT

As so ψ(1)=1, we see that these conditions are satisfied if we take

Equation

Taking correspondingly λ it follows that Equation whenever z ∈ Q0T,τ.

Now we prove that the following holds whenever Equation

Equation

This completes the proof of Theorem 2.1.

Proof of theorem 2.2

After extending u(x,t) as in the above we choose

R=dist0,∂Ω) and define

Equation

We let Z=Ω-0 × (0) then

Equation for every r ∈ (0,R),

Qλψr (r)(z)∩Q0T.

Whenever zZ. Consider Equation and defineEquation If r>R/2, then

Equation for every r ∈ (0,R).

In the final

λ ¯ = max{4λψr ), s·r /ψ ¯ (¯r )} 4max{λ, sR /ψ (R)} = c·λ, implies that

Equation for every r ∈ [0,r¯].

Whenever Equation.

This completes the proof of Theorem 2.2.

Proof of theorem 2.3

Let u(x,t) weakly solve of eqn. (1.1) in QT \E and assume that Equation and Hψ(·)(E)=0. Equation be arbitrary spacetime smooth cylinders. Our only need to prove the conclusion in Q1T since the one of being a weak solution is a local property. By the assumption

Equation there exists M>0 such that

Equation (2.5)

If we using the existence result, then see that there exist a unique solution v(x,t) of problem

Equation (2.6)

Equation

Let μ be the nonnegative Riesz measure associated to v(x,t). Note that from existence μ follows v(x,t) is a supersolution [7]. Let F={(x,t) ∈ Q1T: v(x,t)=u(x,t)}. Now prove that the support of μ is contained in FE. For these is sufficient to show that v(x,t) is a weak solution to (2.6) in Q1T \(FE). We already know that (2.6) satisfy in Q1T \F and it therefore remains to show that (2.6) satisfy in Equation. To this aim, we show that if Equation is a cylinder and Equation is a weak solution toEquation witk α=u on Γ(QT ), then actually v must coincide with α (x,t)in the(QT ) . Note that such a unique solution α(x,t) exists. We immediately see by the comparison principle that vα in Equation, because v(x,t) is a weak supersolution. To show that vα we instead argue as follows: since u(x,t) ≤ v(x,t), we also have u(x,t) ≤ α(x,t) on Γ QT and as u(x,t) solves eqn. (1.1) in QT , the comparison principle holdsu(x,t)≤ α (x,t)in. We thus conclude that v(x,t) ≤ α(x,t) on EquationF . Therefore v(x,t)=α(x,t) and consequently also eqn. (2.6) yields in Equation. This completes the proof that support of μ is contained in FE.

Later using Theorem 2.1 and a covering argument we can conclude that there exists C depending only on Equation such that

Equation (2.7).

Whenever Equation. Consider concentric cylindersEquation. In the following we will use the short notation Equation be such 0 and ϕ ≡ 1 on Q˜τ. Let k=supv(x,t). Using eqn. (2.6) we have

Equation (2.8)

For the nonnegative weak sub solution kv(x,t) we see that

Equation

for some const c=c(n,p,ν,L) ≥ 1. By eqn. (2.7)

Equation

and putting the estimates (2.8) we obtain that

Equation (2.9)

Here we also used the estimate Equation for τ ≤ 1. Now we consider cylinder Equation. We will prove that Equation. We first note using eqn. (2.9) we have

Equation (2.10)

Whenever Equation. Since Hψ(·)(E)=0 we obtain for ε>0 and δ>0 given (to be taken smaller that dist Equation, then there exists a countable family

Equation

of cylinders with 0 < τi < δ,i=1,2,..., such that Equation and

E ∩ Q3T ⊂ [Q˜ψτi(τi) and Xτinψ(τi)<ε. (2.11)

Later using eqn. (2.10) we is obtain

Equation (2.12)

proving that Equation. The fact that both Q2T and Q3T are arbitrary, we can conclude that Equation. Thus v(x,t) is a solution in Q1T . Finally applying the above argument with u(x,t) replaced by −u(x,t) we deduce that there exist two solutions v1(x,t) and v2(x,t) i.e., eqn. (2.6) for v1 equal to eqn. (2.6) for v2. Such that v1(x,t) ≤ u(x,t) ≤ v2(x,t) and v1(x,t) = v2(x,t) on Γ(Q1T ). It follows that v1=v2=u. Theorem is proof.

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