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In this paper, we obtain the existence result of viscosity solutions to the initial and boundary value problem for a nonlinear degenerate parabolic inhomogeneous equation of the form , where denotes infinity Laplacian given by .

In this paper, we consider the nonlinear degenerate parabolic inhomogeneous equation involving infinity La- place

where

denotes the 3-homogeneous infinity Laplacian. We want to establish the existence result of viscosity solutions to the initial and Dirichlet boundary problem.

The homogeneous infinity Laplace equation

They proved the existence and uniqueness for both Dirichlet and Cauchy problems, established interior and boundary Lipschitz estimates and a Harnack inequality, and also provided numerous explicit solutions. Due to the degeneracy and the singularity of the Equation (2), they introduced the approximating equations to obtain the existence result with the aid of the uniform continuity estimates. And in [

is related to game theory named tug-of-war [

They also introduced the corresponding approximating equations and got the uniform continuity estimates of approximate solutions by the barrier function arguments. By this approximate procedure, the existence of the solutions was obtained. They also proved the uniqueness and the asymptotic behavior of the viscosity solutions. In [

with

By the density-to-pressure transformation, they transformed the Equation (4) into a new equation, then the existence, uniqueness and asymptotic behavior etc. were obtained.

In this article, we are interested in the parabolic version of the infinity Laplacian here. We think that Equation (1) is interesting, because it not only is degenerate, but also has many applications in image processing and optimal transportation etc. The parabolic equation involving infinity Laplacian operator has received a lot of attention in the last decade, notably due to its application to image processing, the main usage being in the reconstructions of damaged digital images [

This paper is organized in the following order. In Section 2, we give the notations, definitions of viscosity solutions related to the Equation (1). In Section 3, we prove our main existence result by approximating pro- cedure.

Throughout of this paper, we use the following notation: If

In the following paper, we adopt the definition of viscosity solutions, (see for example [

Definition 2.1. Suppose that

then we say that

Similarly,

then we say that

If

In this section we will prove the existence of viscosity solutions to (1) with the initial and boundary data

Theorem 3.1. Let

in

We use the approximate procedure, cf, [

where

with

Theorem 3.2. (Boundary regularity at

Then there exists a constant

Moreover, if

Proof. Step 1. Suppose first that

where

if

Clearly,

if

Thus, by the classical comparison principle, we obtain

for every

we obtain the symmetric inequality, and hence the Lipschitz estimate

for

Step 2. Suppose now that

a point

It is easy to check that

Thus if

where

with this inequality it is straightforward to complete the proof. □

The full Lipschitz estimate in time now follows easily with the aid of the comparison principle and the fact that the Equation (7) is translation invariant.

Corollary 3.3. (Lipschitz regularity in time) If

Theorem 3.2, then there exists a constant

for all

Proof. Let

and hence if

by the classical comparison principle and Theorem 3.2. This implies the Lipschitz estimate asserted above, and the proof for the case when

Theorem 3.4. (Hölder regularity at the lateral boundary) Let

Then for each

for all

Proof. Step 1. For every

where

If

Therefore

if

We have shown that

Step 2. Let

provided

Case 2. If

provided

Step 3. To prove

Case 1. If

if

Case 2. If

if

Step 4. In conclusion, we have shown that

Therefore, we have

for

we get the symmetric inequality. This finishes the proof. □

Due to the translation invariant of the equation and the comparison principle, we can extend the Hölder estimate to the interior of the domain, cf. [

Corollary 3.5. (Hölder regularity in space) Let

Then there exists

for all

Proof. Step 1. For fixed

By Theorem 3.4 we have that

Step 2. When

This finishes the proof. □

The following theorem shows that one can obtain the Lipschitz estimate when one remove the Laplacian term from the equation, cf. [

Theorem 3.1 follows now easily from Theorem 3.2 and 3.3 and the stability properties of viscosity solutions.

Proof. (Proof of Theorem 3.1) If

Corollaries 3.3 and 3.5 and the comparison principle imply that the family of functions

The existence for a general continuous data

The author would like to thank the anonymous referee for some valuable suggestions.

This work is supported by the National Natural Science Foundation of China, No.11171153.