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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+994506728756

Tel:[email protected]

E-mail:

**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.

**Visit for more related articles at** Journal of Applied & Computational Mathematics

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.

Degenerate; Nonlinear parabolic equations; Regularity; Removability

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

u_{t}−div(ω(x)|Du|^{p−2}Du)=0 (1.1)

u|_{Γ(QT})=h, (1.2)

where Γ(Q_{T})=(Ω¯ × {0}) ∪ (∂Ω × [0,T]) denote the parabolic boundary
of Q_{T}, h : Q_{T}→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 weighted space, where norm following:

where the parabolic metric is defined as

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 Q^{0}_{T} ⊂ Q_{T} be a bounded space time cylinder such that (interior
regularity)

(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 estimates on for every Ω, ⊂ Ω. We doing
remark that in this case is not a compact subset of Q_{T} .

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)

(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 space. In the space

the norm denote the space of equivalence classes of
functions f with distributional gradient Df, both of which are p^{th} power
integral on Q_{T} . Let

be the norm in

Given t_{1}<t_{2} we denote by the space of functions
such that for almost every t, t_{1} ≤ t ≤ t_{2} the function

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

Q_{T} ⊂ R^{n+1} if whenever Q^{0}_{T}=Ω^{0} × (t_{1},t_{2}) ⊂ Q_{T} with Ω^{0} ⊂ Ω ⊂ R^{n} and
t_{1}<t_{2} then and

(2.3)

for all nonnegative .

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

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 . We also
define Hausdorff measure as follows. We let for fixed δ,0< δ<r_{0} and be a family of cylinders such that and 0 < r_{i} < δ for i=1,2,..

Using this notation we let

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

**Theorem 2.3**

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

Assume also that H^{ψ(·)}(E)=0. Then the set E is removable, i.e., u(x,t)
can be extended to be a weak solution in Q_{T} .

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} ⊂ Q_{T} such that Q^{0}_{T} ∩ Γ(Q_{T} )=∅.

We define function

Then . Let ¯*u* be the unique solution to

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,*diam*Ω,*T*^{1/2}}. As clearly

*T* ≤ (*ψ*(*R*))^{2−p} *R ^{p}* ≤

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

(2.4)

We may assume that , where Ω^{0} ⊂ Ω and *τ*>0. We
let *R* be

a number subject to the restrictions

*R* ≤ *dist* (Ω^{0},∂Ω
),*τ* ≥ *R ^{p}* max{osch,

Q_{T}

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

Taking correspondingly λ it follows that whenever z ∈ *Q*^{0}* _{T,τ}*.

Now we prove that the following holds whenever

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*=*dist*(Ω^{0},∂Ω) and define

We let Z=Ω^{-0} × (0) then

for every *r* ∈ (0,*R*),

*Qλψr* (*r*)(*z*)∩*Q*0*T*.

Whenever *z* ∈ *Z*. Consider and define If *r*>*R*/2, then

for every *r* ∈ (0,*R*).

In the final

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

for every *r* ∈ [0,*r*¯].

Whenever .

This completes the proof of Theorem 2.2.

**Proof of theorem 2.3**

Let *u*(*x*,*t*) weakly solve of eqn. (1.1) in *Q _{T}* \

there exists *M*>0 such that

(2.5)

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

(2.6)

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*) ∈ *Q*^{1}* _{T}*:

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

(2.7).

Whenever . Consider concentric cylinders. In the following we will use the short
notation be such 0 and *ϕ* ≡ 1 on *Q*˜* _{τ}*. Let

(2.8)

For the nonnegative weak sub solution *k*−*v*(*x*,*t*) we see that

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

and putting the estimates (2.8) we obtain that

(2.9)

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

(2.10)

Whenever . Since *H ^{ψ}*

of cylinders with 0 < *τ _{i}* <

*E* ∩ Q3*T* ⊂ [Q˜*ψτi*(*τi*) and X*τinψ*(*τi*)<*ε*. (2.11)

Later using eqn. (2.10) we is obtain

(2.12)

proving that . The fact that both Q^{2}* _{T}* and Q

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