On the behavior in time of the solutions to total variation flow

Maria Michaela Porzio; Giuseppe Riey

doi:10.1515/acv-2024-0033

Artikel Open Access

On the behavior in time of the solutions to total variation flow

Maria Michaela Porzio und Giuseppe Riey

Veröffentlicht/Copyright: 2. Oktober 2024

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Abstract

In this paper we study the regularity and the behavior in time of the solutions to the Dirichlet problem for the total variation flow in a bounded domain. We show that the regularity of the initial datum influences the behavior and can produce extinction in finite time and “immediately boundedness of the solutions”.

Keywords: Decay estimates; singular diffusion equations; 1-Laplacian; Dirichlet problem; total variation flow

MSC 2020: 35A01; 35A02; 35K55; 35K20; 35K67; 35D30; 35K92

1 Introduction

The problem considered here is the following:

(1.1) { ∂ ⁡ u ∂ ⁡ t - div ⁡ ( ∇ ⁡ u | ∇ ⁡ u | ) = 0 in ⁢ Ω T , u = 0 on ⁢ ∂ ⁡ Ω × ( 0 , T ) , u ⁢ ( x , 0 ) = u 0 ⁢ ( x ) on ⁢ Ω ,

where T > 0 , Ω is an open bounded set of ℝ N ( N ≥ 2 ) with Lipschitz boundary, Ω T = Ω × ( 0 , T ) and div and ∇ are related to the differentiation only with respect to the spatial variable x ∈ Ω .

The operator div ⁡ ( ∇ ⁡ u | ∇ ⁡ u | ) in (1.1) is usually called “1-Laplace operator” and denoted by Δ 1 ⁢ u , because it can be formally seen as the limit as p goes to 1 of the p-Laplace operator Δ p ⁢ u = div ⁡ ( | ∇ ⁡ u | p - 2 ⁢ ∇ ⁡ u ) .

Equation (1.1) can be regarded as the gradient flow associated to the so called Total Variation functional

F ⁢ ( u ) = ∫ Ω | ∇ ⁡ u | ⁢ 𝑑 x ,

which has achieved great interest on last past years especially because of its applications to image processing (see for instance [8, 12, 38, 39]). In order to study properties of functionals with linear growth like F, for both theoretical and applied reasons, the natural framework is the space of functions with bounded variation, BV ⁢ ( Ω ) in short: from the mathematical point of view it is useful to resort to BV-functions because the Sobolev space W 1 , 1 ⁢ ( Ω ) is not reflexive (and hence compactness properties cannot achieved in order to apply the direct methods of the Calculus of Variations); from the applied point of view, differently from functions belonging to Sobolev spaces, BV-functions are allowed to be discontinuous along sets of co-dimension one and, for example in the case of images in dimension two, such discontinuities can model the contours of the picture to be treated.

In order to give sense to (1.1), several notions of solutions where given in literature. The first one can be found in [2, 3, 4], where existence results and other properties are studied by means of semigroup theory. Moreover, in order to clarify the meaning of the quantity ∇ ⁡ u | ∇ ⁡ u | when the distributional gradient of u is a measure, they use the tools given by Anzellotti in [5] for a suitable definition of pairings between vector fields and measures. More recently another approach (of variational type) was developed in [11, 12] without using the Anzellotti’s theory. In order to handle also the inhomogeneous case, a further technique was used in [41] (with some improvements given in [27]), where the solution for problem (1.1) is shown to be unique and achieved as limit for p → 1 + of solutions of the corresponding parabolic problems involving Δ p ⁢ u instead of Δ 1 ⁢ u . Also in [27, 41] a central role is played by the Anzellotti’s theory and the tools developed there give a further strategy to handle the homogeneous case. These techniques have been recently applied also to elliptic problems involving the 1-Laplace operator (see for instance [17, 26] and the references therein).

Just this last approach will be useful to get the results in the present paper. We point out that although several results are known about the homogeneous case (1.1), there are still some interesting open problems. For example, it is well known that, if the initial datum u 0 ∈ L ∞ ⁢ ( Ω ) , then there exists (and is also unique) a solution u of (1.1) belonging to L ∞ ⁢ ( Ω T ) (see [3] and the references therein). What happens if u 0 ∉ L ∞ ⁢ ( Ω ) ? What is in this case the regularity of the solution? What is the influence of the initial datum u 0 on the regularity of the solutions?

We recall that in the case of the solutions u p of the p-Laplacian equation

(1.2) { ∂ ⁡ u p ∂ ⁡ t - div ⁡ ( | ∇ ⁡ u p | p - 2 ⁢ ∇ ⁡ u p ) = 0 in ⁢ Ω T , u p = 0 on ⁢ ∂ ⁡ Ω × ( 0 , T ) , u p ⁢ ( x , 0 ) = u 0 ⁢ ( x ) on ⁢ Ω ,

if the initial datum u 0 ⁢ ( x ) belongs to L r 0 ⁢ ( Ω ) with r 0 ≥ 1 satisfying

(1.3) p > 2 ⁢ N N + r 0 ,

then the solutions u become immediately bounded and satisfy the following L ∞ -bound:

(1.4) ∥ u p ⁢ ( t ) ∥ L ∞ ⁢ ( Ω ) ≤ C 1 ⁢ ∥ u 0 ∥ L r 0 ⁢ ( Ω ) p ⁢ r 0 N ⁢ ( p - 2 ) + p ⁢ r 0 t N N ⁢ ( p - 2 ) + p ⁢ r 0

(see [43, 23, 20, 19, 42, 13, 32] and [24] if N = 1 ). Estimates like (1.4) are known in literature as decay estimates or ultracontractive estimates and they are known to be true not only for the p-Laplacian equation but also for other evolution problems like the porous media equation, the fast diffusion equation, the doubly nonlinear equations and other type of nonlinear problems singular or degenerate (see [7, 9, 22, 43, 10, 40, 13, 32, 18, 31, 15, 21, 16, 30, 42] and the references therein).

The interest in this kind of estimates is due to the fact that they reveal a very strong regularizing effect of these problems, that is: as soon as t > 0 , the solutions immediately become bounded even if their initial data are unbounded. Moreover, they describe the behavior in time of the solutions for every value of t. In particular, they imply that for t large the solutions decay (in the L ∞ -norm) being the right-hand side of (1.4) a function converging to zero as t → + ∞ . Furthermore, these estimates describe also the blow-up, as t → 0 , of the L ∞ -norm of u ⁢ ( t ) (that occurs in the case of unbounded initial data) and allow to conclude that the blow-up is “controlled” by the power t - N N ⁢ ( p - 2 ) + p ⁢ r 0 .

Hence, another question (strictly related to the previous one) is the following: is it true that also the solution of (1.1), even if u 0 ∉ L ∞ ⁢ ( Ω ) , as soon as t > 0 becomes immediately bounded and satisfies a decay estimate? And in the affirmative case, under which assumption on u 0 does this regularizing phenomenon appear? We recall that in the case of the p-Laplacian equation, condition (1.3) is a necessary condition to have this instantaneous boundedness of u (see [20]).

Another interesting question is related to the following result: it is known that if the initial datum is bounded the solution of (1.1) extinguishes in finite time, i.e. there exists a constant T 0 > 0 such that it results

(1.5) u ⁢ ( x , t ) = 0 a.e. ⁢ x ∈ Ω and for every ⁢ t > T 0 .

Moreover, the following estimate of T 0 (in dependence of u 0 ) holds true:

(1.6) T 0 ≤ d ⁢ ( Ω ) ⁢ ∥ u 0 ∥ L ∞ ⁢ ( Ω ) N ,

where d ⁢ ( Ω ) is the smallest radius of the ball containing Ω (see [3, Corollary 1]).

Is it possible that the solution of (1.1) extinguishes in finite time even if u 0 ∉ L ∞ ⁢ ( Ω ) ? And in the affirmative case, is it possible to estimate the extinction time T 0 such that (1.5) holds true?

Notice that in the case of u 0 ∈ L ∞ ⁢ ( Ω ) , the above estimate (1.6) of T 0 depends on the L ∞ -norm of u 0 , and consequently it cannot help in the case of unbounded initial data.

In this paper we have answered to the previous questions proving that if u 0 ∈ L r 0 ⁢ ( Ω ) with r 0 ≥ 2 satisfying

(1.7) r 0 > N ,

then the solutions u of (1.1) become immediately bounded and satisfy the following L ∞ -bound:

(1.8) ∥ u ⁢ ( t ) ∥ L ∞ ⁢ ( Ω ) ≤ C 1 ⁢ ∥ u 0 ∥ L r 0 ⁢ ( Ω ) r 0 r 0 - N t N r 0 - N .

Moreover, we prove that extinction in finite time occurs as soon as u 0 ∈ L N ⁢ ( Ω ) and the following estimate on the extinction time T 0 holds true:

T 0 ≤ ∥ u 0 ∥ L N ⁢ ( Ω ) .

Indeed, we prove that also in this case the solution u immediately becomes bounded.

Finally, we discuss also what happens if u 0 ∉ L N ⁢ ( Ω ) , that is the case when we cannot expect L ∞ -regularization and the decay of the solution in the L ∞ -norm and we prove that also in this case the solution decays but this happens in a L r -norm (for suitable r related to the summability of the datum u 0 ).

The paper runs as follows. In Section 2 we collect some preliminary results and in Section 3 we state our mains results, which will be proved in Section 4.

2 Notation and preliminary results

For Ω ⊂ ℝ N , u ∈ L 1 ⁢ ( Ω ) is said of “bounded variation” if its distributional gradient is a vectorial Radon measure (denoted by Du) with finite total variation (denoted by | D ⁢ u | ) and the space of such functions is denoted by BV ⁢ ( Ω ) (for details about BV-functions see for instance [1]).

For u ∈ BV ⁢ ( Ω ) , ∇ ⁡ u is used to represent the derivative of Radon–Nikodym of Du with respect to the Lebesgue measure λ. For u ∈ W 1 , 1 ⁢ ( Ω ) we just have D ⁢ u = ∇ ⁡ u ⁢ λ , namely the distributional derivative of a Sobolev function can be seen as a measure absolutely continuous with respect to λ.

In order to study the minimizing properties of a functional defined on W 1 , 1 ⁢ ( Ω ) , it is needed to extend it to BV ⁢ ( Ω ) and then considering its lower semicontinuous envelope, usually called “relaxed functional” (see for instance [1]). For

F ⁢ ( u ) = ∫ Ω | ∇ ⁡ u | ⁢ 𝑑 λ ,

the relaxed of F is

F ¯ ⁢ ( u ) = | D ⁢ u | ⁢ ( Ω ) = ∫ Ω d ⁢ | D ⁢ u | .

Therefore, if we formally compute the Euler equation of F, we get the operator Δ 1 appearing in (1.1), which in sense represents the first variation restricted to Sobolev functions and it does not take into account the information in the other part of the distributional gradient singular with respect to λ and related in particular to jumps and Cantor parts. A rigorous way to derive the Euler equation for functionals defined in BV ⁢ ( Ω ) was given in [6] and an example of application of it to a functional with linear growth (useful in image processing) can be found in [28]. However in the present paper we do not follow this last approach and we resort to the notion of solution performed in [41]), which allows us to better achieve some qualitative properties for the parabolic problem (1.1).

For this purpose we recall the following tools.

Given z ∈ L ∞ ⁢ ( Ω ) with div ⁡ z ∈ L 2 ⁢ ( Ω ) , it is possible to define (see [5]) a weak trace on ∂ ⁡ Ω of the normal component of z, denoted by [ z , ν ] , belonging to L ∞ ⁢ ( Ω ) and such that, for w ∈ W 1 , 1 ⁢ ( Ω ) ∩ L 2 ⁢ ( Ω ) , there holds

∫ ∂ ⁡ Ω w ⁢ [ z , ν ] ⁢ 𝑑 ℋ N - 1 = ∫ Ω w ⁢ div ⁡ z ⁢ d ⁢ x + ∫ Ω z ⋅ ∇ ⁡ w ⁢ d ⁢ x ,

where ℋ N - 1 stands for the ( N - 1 ) -dimensional Hausdorff measure.

Moreover, if v ∈ BV ⁢ ( Ω ) ∩ L 2 ⁢ ( Ω ) , it is possible to define the distribution ( z , D ⁢ v ) : C 0 ∞ ⁢ ( Ω ) → ℝ as follows:

〈 ( z , D ⁢ v ) , φ 〉 = - ∫ Ω v ⁢ z ⋅ ∇ ⁡ φ ⁢ d ⁢ x - ∫ Ω v ⁢ φ ⁢ div ⁡ z ⁢ d ⁢ x , φ ∈ C 0 ∞ ⁢ ( Ω ) ,

and to show that ( z , D ⁢ v ) is a Radon measure^[1] with finite total variation satisfying

∫ Ω v ⁢ div ⁡ z ⁢ d ⁢ x + ∫ Ω ( z , D ⁢ v ) = ∫ Ω [ z , ν ] ⁢ v ⁢ 𝑑 ℋ N - 1 .

In [2] these results are generalized to z ∈ L ∞ ⁢ ( Ω ) with div ⁡ z ∈ L 1 ⁢ ( Ω ) and in [29] with div ⁡ z belonging to the predual of BV ⁢ ( Ω ) .

Given a function u = u ⁢ ( x , t ) defined on Ω T , we see it as a map from [ 0 , T ] to a suitable space X of functions defined on Ω and we will write u ′ for the derivative of u with respect to the variable t.

Given a Banach space X, X * stands for its dual. We recall that BV ⁢ ( Ω ) endowed with the norm

∥ u ∥ BV ⁢ ( Ω ) = | D ⁢ u | ⁢ ( Ω ) + ∥ u ∥ L 1 ⁢ ( Ω )

is a Banach space.

We say that u belongs to L loc r ⁢ ( ( 0 , + ∞ ) ; X ) if u belongs to L r ⁢ ( ( 0 , s ) ; X ) for every s > 0 .

A function u : [ 0 , T ] → BV ⁢ ( Ω ) is said to be “weakly measurable” if the function

f ⁢ ( t ) = 〈 ϕ , u ⁢ ( t ) 〉

is measurable for every ϕ ∈ BV ⁢ ( Ω ) * .

We denote by L w 1 ⁢ ( ( 0 , T ) ; BV ⁢ ( Ω ) ) the space of all weakly measurable maps u : [ 0 , T ] → BV ⁢ ( Ω ) such that

∫ 0 T ∥ u ⁢ ( t ) ∥ BV ⁢ ( Ω ) < + ∞ .

Definition 2.1.

We say that f ∈ L 1 ⁢ ( ( 0 , T ) ; BV ⁢ ( Ω ) ) ∩ L ∞ ⁢ ( ( 0 , T ) ; L 2 ⁢ ( Ω ) ) has a “weak derivative”

g ∈ L w 1 ⁢ ( ( 0 , T ) ; BV ⁢ ( Ω ) ) ∩ L ∞ ⁢ ( Ω T )

f ⁢ ( t ) = ∫ 0 t g ⁢ ( s ) ⁢ 𝑑 s ,

where the integral is defined as a Pettis integral.

Definition 2.2.

Given u ∈ C ⁢ ( [ 0 , T ] ; L 2 ⁢ ( Ω ) ) ∩ L 1 ⁢ ( ( 0 , T ) ; BV ⁢ ( Ω ) ) , we say that u ′ ∈ L 1 ⁢ ( ( 0 , T ) ; BV ⁢ ( Ω ) ) * is the “time derivative” of u if

∫ 0 T 〈 u ′ ⁢ ( t ) , f ⁢ ( t ) 〉 ⁢ 𝑑 t = - ∫ 0 T ∫ Ω u ⁢ ( s , x ) ⁢ g ⁢ ( s , x ) ⁢ 𝑑 x ⁢ 𝑑 s

for every f ∈ L 1 ⁢ ( ( 0 , T ) ; BV ⁢ ( Ω ) ) ∩ L ∞ ⁢ ( ( 0 , T ) ; L 2 ⁢ ( Ω ) ) with weak derivative g ∈ L w 1 ⁢ ( ( 0 , T ) ; BV ⁢ ( Ω ) ) ∩ L ∞ ⁢ ( Ω T ) and compact support in time.

We can now recall the meaning of solution and global solution of problem (1.1).

Definition 2.3.

Let u 0 ∈ L 2 ⁢ ( Ω ) . We say that u ∈ L 1 ⁢ ( ( 0 , T ) ; BV ⁢ ( Ω ) ) ∩ C ⁢ ( [ 0 , T ] ; L 2 ⁢ ( Ω ) ) is a solution of (1.1) if there exist z ∈ L ∞ ⁢ ( Ω T ) and u ′ ∈ L 1 ⁢ ( ( 0 , T ) ; BV ⁢ ( Ω ) ) * such that u ′ is the time derivative of u and for a.e. t ∈ [ 0 , T ] there holds:

(2.1) { u ′ ⁢ ( t ) = div ⁡ z ⁢ ( t ) in ⁢ 𝒟 ′ ⁢ ( Ω ) , ( z ⁢ ( t ) , D ⁢ u ⁢ ( t ) ) = | D ⁢ u ⁢ ( t ) | , ∈ sign ⁡ ( - u ⁢ ( t ) ) ℋ N - 1 ⁢ a.e. on ⁢ ∂ ⁡ Ω , u ⁢ ( x , 0 ) = u 0 ⁢ ( x ) in ⁢ Ω ,

where sign ⁡ ( s ) is the multivalue function

sign ⁡ ( s ) = { 1 if ⁢ s > 0 , if ⁢ s = 0 , - 1 if ⁢ s < 0 .

Remark 2.1.

The third condition in (2.1) generalizes to the BV framework the Dirichlet boundary condition in (1.1). In particular, we recall that, when we deal with BV functions (which are allowed to jump possibly also on ∂ ⁡ Ω ), the Dirichlet conditions “ u = 0 on ∂ ⁡ Ω ” has to be thought as an external constraint, namely the external trace of u on ∂ ⁡ Ω is imposed to be 0 and the internal trace of u is not necessarily equal to 0. We recall that D ⁢ u | D ⁢ u | (thought as the density obtained computing the derivative of Radon–Nikodym of Du with respect to | D ⁢ u | ), when u is a Sobolev function, coincides with the normal vector to the level curves of u pointing in the increasing direction of u, while, when u is a BV function with for instance a jump discontinuity along a curve, it coincides with the normal to the jump pointing in the direction of bigger trace. Moreover, we recall that the vector field z play the role of D ⁢ u | D ⁢ u | and that the quantity [ z , ν ] generalizes the normal component of z, namely the scalar product of z with ν the external normal to ∂ ⁡ Ω . Therefore, if the internal trace of u on ∂ ⁡ Ω is for instance positive, we expect [ z , ν ] to be negative. If the internal trace of u is equal to 0, [ z , ν ] is allowed to be positive or negative and so the third condition in (2.1) is explained in order to ensure the regularity given by the Anzellotti’s theory [5].

Definition 2.4.

We say that u ∈ L loc 1 ⁢ ( ( 0 , + ∞ ) ; BV ⁢ ( Ω ) ) ∩ C ⁢ ( [ 0 , + ∞ ) ; L 2 ⁢ ( Ω ) ) is a global solution of (1.1), or equivalently, u is a solution of the problem

(2.2) { ∂ ⁡ u ∂ ⁡ t - div ⁡ ( ∇ ⁡ u | ∇ ⁡ u | ) = 0 in ⁢ Ω ∞ ≡ Ω × ( 0 , + ∞ ) , u = 0 on ⁢ ∂ ⁡ Ω × ( 0 , + ∞ ) , u ⁢ ( x , 0 ) = u 0 ⁢ ( x ) on ⁢ Ω ,

if it is a solution of (1.1) for every T > 0 (i.e. if it satisfies (2.1) for every T > 0 ).

In the following theorem we summarize the results in [41, Theorem 4.1 and Proposition 5.2].

Theorem 2.1.

Let u 0 ∈ L 2 ⁢ ( Ω ) . Then there exists a unique solution u of (1.1) satisfying u ′ ⁢ ( t ) ∈ L 2 ⁢ ( Ω ) for a.e. t ∈ [ 0 , T ] and

(2.3) max t ∈ [ 0 , T ] ⁢ ∥ u ⁢ ( t ) ∥ L 2 ⁢ ( Ω ) ≤ ∥ u 0 ∥ L 2 ⁢ ( Ω ) .

Moreover, u can be obtained as limit (as p → 1 + ) a.e. in Ω T and in L 1 ⁢ ( Ω T ) of functions u p ∈ C ⁢ ( [ 0 , T ] ; L 2 ⁢ ( Ω ) ) ∩ L p ⁢ ( 0 , T ; W 0 1 , p ⁢ ( Ω ) ) which are distributional solutions of the following nonlinear singular parabolic problems:

(2.4) { ∂ ⁡ u p ∂ ⁡ t - div ⁡ ( | ∇ ⁡ u p | p - 2 ⁢ ∇ ⁡ u p ) = 0 in ⁢ Ω T , u p = 0 on ⁢ ∂ ⁡ Ω × ( 0 , T ) , u p ⁢ ( x , 0 ) = u 0 ⁢ ( x ) on ⁢ Ω .

An immediate consequence of the previous theorem is the following result.

Theorem 2.2.

Let u 0 ∈ L 2 ⁢ ( Ω ) . Then there exists a unique global solution u of (1.1) satisfying u ′ ⁢ ( t ) ∈ L 2 ⁢ ( Ω ) for a.e. t > 0 and

(2.5) max t > 0 ⁢ ∥ u ⁢ ( t ) ∥ L 2 ⁢ ( Ω ) ≤ ∥ u 0 ∥ L 2 ⁢ ( Ω ) .

Moreover, for every T > 0 , u can be obtained as limit (as p → 1 + ) a.e. in Ω T and in L 1 ⁢ ( Ω T ) of the distributional solutions u p ∈ C ⁢ ( [ 0 , T ] ; L 2 ⁢ ( Ω ) ) ∩ L p ⁢ ( 0 , T ; W 0 1 , p ⁢ ( Ω ) ) of (2.4).

Proof.

For every T > 0 let us denote u ( T ) the unique solution of (1.1) (given by Theorem 2.1) which satisfies ( u ( T ) ) ′ ∈ L 2 ⁢ ( Ω ) for a.e. t ∈ [ 0 , T ] and verifying estimate (2.3). Hence, for every T 0 > 0 , it results

u ( T ) = u ( T 0 ) in ⁢ Ω × ( 0 , min ⁡ { T , T 0 } ) .

Hence, for every ( x , t ) ∈ Ω ∞ the following function

u ⁢ ( x , t ) ≡ u ( T ) ⁢ ( x , t ) ,

with T > t arbitrarily chosen, is well defined and satisfies all the statements of the theorem. ∎

For the convenience of the readers, we conclude this section recalling a result proved in [32], that shows that it is possible to derive L ∞ -estimates for a function u simply by proving that such a function satisfies suitable integral estimates.

Theorem 2.3 ([32, Theorem 2.1]).

Let Ω be an open set of R N (not necessary bounded), N ≥ 1 , 0 < T ≤ + ∞ and u a function in C ⁢ ( ( 0 , T ) ; L r ⁢ ( Ω ) ) ∩ L b ⁢ ( 0 , T ; L q ⁢ ( Ω ) ) ∩ C ⁢ ( [ 0 , T ) ; L r 0 ⁢ ( Ω ) ) , where

(2.6) 1 ≤ r 0 < r < q ≤ + ∞ , b 0 < b < q , b 0 = r - r 0 1 - r 0 q .

Assume that u satisfies the following integral estimates for every k > 0 :

(2.7) ∫ Ω | G k ⁢ ( u ) | r ⁢ ( t 2 ) ⁢ 𝑑 x - ∫ Ω | G k ⁢ ( u ) | r ⁢ ( t 1 ) ⁢ 𝑑 x + c 1 ⁢ ∫ t 1 t 2 ∥ G k ⁢ ( u ) ⁢ ( τ ) ∥ L q ⁢ ( Ω ) b ⁢ 𝑑 τ ≤ 0 for all ⁢ 0 < t 1 < t 2 < T

and

(2.8) ∥ G k ⁢ ( u ) ⁢ ( t ) ∥ L r 0 ⁢ ( Ω ) ≤ c 2 ⁢ ∥ G k ⁢ ( u ) ⁢ ( t 0 ) ∥ L r 0 ⁢ ( Ω ) for all ⁢ 0 ≤ t 0 < t < T ,

where c 1 and c 2 are positive constants independent of k,

(2.9) G k ⁢ ( s ) = ( | s | - k ) + ⁢ sign ⁡ ( s ) , s ∈ ℝ ,

and

(2.10) u 0 ≡ u ⁢ ( x , 0 ) ∈ L r 0 ⁢ ( Ω ) .

Then there exists a positive constant C 1 , depending only on N, c 1 , c 2 , r, r 0 , q and b, such that

(2.11) ∥ u ⁢ ( t ) ∥ L ∞ ⁢ ( Ω ) ≤ C 1 ⁢ ∥ u 0 ∥ L r 0 ⁢ ( Ω ) h 0 t h 1 , t ∈ ( 0 , T ) ,

where

(2.12) h 1 = 1 b - ( r - r 0 ) - r 0 ⁢ b q , h 0 = h 1 ⁢ ( 1 - b q ) ⁢ r 0 .

We point out that the following estimate for C 1 holds true (see [32, formula (4.11)]):

(2.13) C 1 = [ ( r - r 0 ) ⁢ h 1 ] h 1 ⁢ 2 4 ⁢ ( 1 + h 1 ) ( r - r 0 ) ⁢ ( 1 - b q ) ⁢ h 1 ⁢ c 2 h 0 + h 1 ⁢ r 0 ⁢ b ⁢ ( 1 - r q ) c 1 h 1 .

For an easy description of this new method to derive decay estimates by integral inequalities see [35], while further developments of this technique can be found in [34] and [36].

3 Statement of results

We have the following results.

Theorem 3.1.

If u 0 ∈ L r 0 ⁢ ( Ω ) with r 0 > N then there exists a global solution u of (1.1) belonging to L ∞ ⁢ ( Ω × ( ε , + ∞ ) ) for every ε > 0 , which satisfies the following estimate for every t > 0 :

(3.1) ∥ u ⁢ ( t ) ∥ L ∞ ⁢ ( Ω ) ≤ C 0 ⁢ ∥ u 0 ∥ L r 0 ⁢ ( Ω ) r 0 r 0 - N t N r 0 - N ,

where C 0 is a constant depending only on r 0 and N (see formula (4.10)). Moreover, for every T > 0 , u is the a.e. limit in Ω T of the solutions u p of (2.4) and is the unique global solution of (1.1) which satisfies u ′ ⁢ ( t ) ∈ L 2 ⁢ ( Ω ) for a.e. t > 0 and which verifies the bound (2.5).

Remark 3.1.

We point out that the previous result affirms that even if the initial datum u 0 is not bounded, if it belongs to L r 0 ⁢ ( Ω ) , with r 0 > N , then the global solution u of (1.1) becomes immediately bounded and satisfies the bound (3.1).

As recalled in the introduction, the solutions u p of the p-Laplacian equations (2.4) satisfy a similar property: if it results

(3.2) p > 2 ⁢ N N + r 0 ,

then the solutions u p become immediately bounded and satisfy the following estimate:

(3.3) ∥ u p ⁢ ( t ) ∥ L ∞ ⁢ ( Ω ) ≤ C 1 ⁢ ∥ u 0 ∥ L r 0 ⁢ ( Ω ) p ⁢ r 0 N ⁢ ( p - 2 ) + p ⁢ r 0 t N N ⁢ ( p - 2 ) + p ⁢ r 0 ,

(see [32] and the references therein). We observe that it results

1 > 2 ⁢ N N + r 0 ⇔ r 0 > N ,

i.e. (3.2) for p → 1 becomes our assumption on the summability coefficient r 0 of the initial datum u 0 .

Moreover, it results

lim p → 1 ⁡ p ⁢ r 0 N ⁢ ( p - 2 ) + p ⁢ r 0 = r 0 r 0 - N , lim p → 1 ⁡ N N ⁢ ( p - 2 ) + p ⁢ r 0 = N r 0 - N ,

i.e., for p → 1 the decay exponents in the right-hand side of (3.3) (which are known to be sharp) become the decay exponents in the right-hand side of (3.1).

We observe that estimate (3.1) implies the following asymptotic behavior:

lim t → + ∞ ⁡ ∥ u ⁢ ( t ) ∥ L ∞ ⁢ ( Ω ) = 0 .

Moreover, estimate (3.1) allows also to describe the possible blow-up of u ⁢ ( t ) as t → 0 . As a matter of fact, if u 0 ∉ L ∞ ⁢ ( Ω ) , then the L ∞ -norm of u ⁢ ( t ) blows up as t → 0 , but, thanks to estimate (3.1) we know that such a blow-up is controlled by the power t - N r 0 - N .

We observe that there exists a solution of (1.1) which becomes “immediately bounded” also if the initial datum belongs to L N ⁢ ( Ω ) . Moreover, if the initial datum is sufficiently regular (for example under the assumption in Theorem 3.1) it is possible to show that there is extinction in finite time. In details we have

Theorem 3.2.

Let u 0 be in L r 0 ⁢ ( Ω ) with r 0 ≥ N . Then there exists a global solution u of (1.1) and positive value T 0 such that

(3.4) u ⁢ ( x , t ) = 0 a.e. in ⁢ Ω × ( T 0 , T ) .

Moreover, the following estimate on the extinction time T 0 holds:

(3.5) T 0 ≤ ∥ u 0 ∥ L N ⁢ ( Ω ) .

Furthermore, for every T > 0 , u is the a.e. limit in Ω T of the solutions u p of (2.4) and is the unique global solution of (1.1) which satisfies u ′ ⁢ ( t ) ∈ L 2 ⁢ ( Ω ) for a.e. t > 0 and which verifies the bound (2.5).

Finally, if r 0 > N , then u coincides with the solution satisfying the statements of Theorem 3.1. If otherwise r 0 = N , then u belongs to L ∞ ⁢ ( Ω × ( ε , + ∞ ) ) for every ε > 0 .

Remark 3.2.

We recall that if u 0 ∈ L ∞ ⁢ ( Ω ) the solution of (1.1) satisfies the following estimate:

∥ u ⁢ ( t ) ∥ L ∞ ⁢ ( Ω ) ≤ N d ⁢ ( Ω ) ⁢ ( d ⁢ ( Ω ) ⁢ ∥ u 0 ∥ L ∞ ⁢ ( Ω ) N - t ) + ,

where d ⁢ ( Ω ) is the smallest radius of the ball containing Ω (see [3, Corollary 1]). An immediate consequence of the previous estimate is that if u 0 ∈ L ∞ ⁢ ( Ω ) then, as recalled in the introduction, the solution of (1.1) extinguishes in finite time and it results

u ⁢ ( x , t ) = 0 for all ⁢ t ≥ d ⁢ ( Ω ) ⁢ ∥ u 0 ∥ L ∞ ⁢ ( Ω ) N .

Hence, the results in Theorem 3.2 can be seen as a further step in the comprehension of the behavior of the solutions of (1.1) since it reveals that the interesting phenomenon of the extinction in finite time occurs also when the initial datum u 0 is unbounded.

We complete the previous results describing what happens when u 0 does not satisfy the assumptions of previous theorems, i.e. when u 0 ∉ L r 0 ⁢ ( Ω ) with r 0 ≥ N . We have chosen to consider directly “the less regular case” of initial datum u 0 ∈ L 2 ⁢ ( Ω ) to evidence what happens. As the following result shows, also in this case the solution u ⁢ ( t ) decays, but since the immediate L ∞ -regularization does not appear in this case if N > 2 (due to the low regularity of the initial datum), the decay holds true in every Lebesgue space L r ⁢ ( Ω ) with r < 2 .

Theorem 3.3.

If u 0 ∈ L 2 ⁢ ( Ω ) , N > 2 , then there exists a global solution u of (1.1) such that for every r ∈ ( 1 , 2 ) the following estimate holds:

(3.6) ∥ u ⁢ ( t ) ∥ L r ⁢ ( Ω ) ≤ C 1 ⁢ ∥ u 0 ∥ L 2 ⁢ ( Ω ) 2 ⁢ ( N - r ) r ⁢ ( N - 2 ) t N ⁢ ( 2 - r ) r ⁢ ( N - 2 ) ,

where C 1 is a constant depending only on r and N (see formula (2.13)). Moreover, u is the unique global solution of (1.1) which satisfies u ′ ⁢ ( t ) ∈ L 2 ⁢ ( Ω ) for a.e. t > 0 and which verifies the bound (2.5).

Remark 3.3.

The restriction N > 2 in the previous result is done only because the case N = 2 and u 0 ∈ L 2 ⁢ ( Ω ) has been already treated in Theorem 3.2 and, as stated above, in such a case the behavior of the solution differs completely from the case described in Theorem 3.3 since an immediate L ∞ -regularization appears being the summability exponent of the initial datum u 0 equal to the dimension N of the space.

Remark 3.4.

We point out that estimate (3.6) implies

lim t → + ∞ ⁡ ∥ u ⁢ ( t ) ∥ L r ⁢ ( Ω ) = 0 .

Remark 3.5.

We recall that in [33] it is proved that if u 0 ∈ L 2 ⁢ ( Ω ) , every solution u p of the p-Laplacian equation (2.4) with p satisfying

(3.7) 1 < p < 2 ⁢ N N + 2

satisfies the following decay estimate for every r ∈ ( 1 , 2 ) ;

(3.8) ∥ u ⁢ ( t ) ∥ L r ⁢ ( Ω ) ≤ C p ⁢ ∥ u 0 ∥ L 2 ⁢ ( Ω ) 2 ⁢ [ 2 ⁢ N - p ⁢ ( N + r ) ] r ⁢ [ 2 ⁢ N - p ⁢ ( N + 2 ) ] t N ⁢ ( 2 - r ) r ⁢ [ 2 ⁢ N - p ⁢ ( N + 2 ) ] .

We point out that it results

lim p → 1 ⁡ 2 ⁢ [ 2 ⁢ N - p ⁢ ( N + r ) ] r ⁢ [ 2 ⁢ N - p ⁢ ( N + 2 ) ] = 2 ⁢ ( N - r ) r ⁢ ( N - 2 ) , lim p → 1 ⁡ N ⁢ ( 2 - r ) r ⁢ [ 2 ⁢ N - p ⁢ ( N + 2 ) ] = N ⁢ ( 2 - r ) r ⁢ ( N - 2 ) ,

i.e. the decay exponents in the right-hand side of (3.8) become the decay exponents the right-hand side of (3.6).

4 Proofs of the results

In this section we prove all the results stated above in Section 3.

Proof of Theorem 3.1.

By assumption, u 0 ∈ L r 0 ⁢ ( Ω ) with r 0 > N and N ≥ 2 . Consequently, u 0 is in L 2 ⁢ ( Ω ) . Let u be the global solution of (1.1) given by Theorem 2.2 constructed for every T > 0 as the a.e. limit in Ω T of u p ∈ C ⁢ ( [ 0 , T ] ; L 2 ⁢ ( Ω ) ) ∩ L p ⁢ ( 0 , T ; W 0 1 , p ⁢ ( Ω ) ) distributional solutions of (2.4). We prove estimate (3.1) by proving L ∞ -estimates for the solutions u p which are “stable” when p → 1 + . We observe that the assumption r 0 > N implies that 2 ⁢ N N + r 0 < 1 and consequently, every p > 1 satisfies also the inequality p > 2 ⁢ N N + r 0 . Thus, applying [33, Theorem 1.3], we deduce that for every arbitrarily fixed T > 0 the function u p belongs to L ∞ ⁢ ( Ω × ( ε , T ) ) for every ε ∈ ( 0 , T ) . Hence, we can choose | G k ⁢ ( u p ) | r - 2 ⁢ G k ⁢ ( u p ) as test function in the weak form of (2.4), where r > r 0 is arbitrarily fixed and G k is the function defined in (2.9). Notice that the assumption r 0 > N implies that r - 2 > 0 . For every 0 < t 1 < t 2 ≤ T we have

(4.1) ∫ Ω | G k ⁢ ( u p ⁢ ( x , t 2 ) ) | r - ∫ Ω | G k ⁢ ( u p ⁢ ( x , t 1 ) ) | r + r ⁢ ( r - 1 ) ⁢ ∫ t 1 t 2 ∫ Ω | ∇ ⁡ G k ⁢ ( u p ) | p ⁢ | G k ⁢ ( u p ) | r - 2 ≤ 0 .

Notice that, using the Sobolev inequality, we obtain

(4.2)

∫ t 1 t 2 ∫ Ω | ∇ ⁡ G k ⁢ ( u p ) | p ⁢ | G k ⁢ ( u p ) | r - 2 = ( p r - 2 + p ) p ⁢ ∫ t 1 t 2 ∫ Ω | ∇ ⁡ [ | G k ⁢ ( u p ) | r - 2 + p p ⁢ sign ⁡ ( u p ) ] | p

≥ ( p r - 2 + p ) p ⁢ C s ⁢ ∫ t 1 t 2 ∥ | G k ⁢ ( u p ) | r - 2 + p p ∥ L p * ⁢ ( Ω ) p ,

where p * = N ⁢ p N - p is the Sobolev exponent and C s = ( N - 1 ) ⁢ p N - p is a (non-optimal) Sobolev constant (see [14]). By (4.1) and (4.2) we conclude that for every 0 < t 1 < t 2 ≤ T the following integral estimate holds true:

(4.3) ∫ Ω | G k ⁢ ( u p ⁢ ( x , t 2 ) ) | r - ∫ Ω | G k ⁢ ( u p ⁢ ( x , t 1 ) ) | r + c 1 ⁢ ∫ t 1 t 2 ∥ G k ⁢ ( u p ) ∥ L q ⁢ ( Ω ) b ≤ 0 ,

where

(4.4) c 1 ≡ r ⁢ ( r - 1 ) ⁢ ( p r - 2 + p ) p ⁢ ( N - 1 ) ⁢ p N - p

and

(4.5) b = r - 2 + p , q = r - 2 + p p ⁢ p * .

Hence the integral estimate (2.7) of Theorem 2.3 is satisfied. We point out that our choice of the exponents in (4.3) satisfy (2.6). Thus to apply Theorem 2.3 it remains to show that also the integral estimate (2.8) holds true. To this end, it is sufficient to choose | G k ⁢ ( u p ) | r 0 - 2 ⁢ G k ⁢ ( u p ) as test function in the weak form of (2.4) and we deduce that (4.1) holds true with r = r 0 which implies that the following estimate is satisfied for every 0 < t 1 < t 2 ≤ T :

(4.6) ∥ G k ⁢ ( u p ⁢ ( x , t 2 ) ) ∥ L r 0 ⁢ ( Ω ) ≤ ∥ G k ⁢ ( u p ⁢ ( x , t 1 ) ) ∥ L r 0 ⁢ ( Ω ) ,

that is, estimate (2.8) with c 2 = 1 . Hence, applying Theorem 2.3, we deduce that (2.11) holds true, i.e.

(4.7) ∥ u p ⁢ ( t ) ∥ L ∞ ⁢ ( Ω ) ≤ C 1 ⁢ ∥ u 0 ∥ L r 0 ⁢ ( Ω ) h 0 t h 1 , t ∈ ( 0 , T ) ,

where

(4.8) h 0 = p ⁢ r 0 N ⁢ ( p - 2 ) + p ⁢ r 0 , h 1 = N N ⁢ ( p - 2 ) + p ⁢ r 0 ,

and C 1 is as in (2.13). Notice that it results

(4.9) lim p → 1 ⁡ h 0 = r 0 r 0 - N , lim p → 1 ⁡ h 1 = N r 0 - N

and

(4.10) lim p → 1 ⁡ C 1 = C 0 ≡ ( ( r - r 0 ) ⁢ N r ⁢ ( r 0 - N ) ) N r 0 - N ⁢ 2 4 ⁢ r 0 r - r 0 .

Thus, recalling that by construction u p converges a.e. in Ω T to u, by (4.7)–(4.10) we can conclude that

(4.11) ∥ u ⁢ ( t ) ∥ L ∞ ⁢ ( Ω ) ≤ C 1 ⁢ ∥ u 0 ∥ L r 0 ⁢ ( Ω ) r 0 r 0 - N t N r 0 - N , t ∈ ( 0 , T ) ,

for every T > 0 and hence (3.1) follows. We point out that the previous estimate implies that u belongs to L ∞ ⁢ ( Ω × ( ε , + ∞ ) ) for every ε > 0 and hence the theorem is proved. ∎

Proof of Theorem 3.2.

Let u be the global solution of (1.1) given by Theorem 2.2 obtained, for every T > 0 , as the a.e. limit in Ω T of the distributional solutions u p ∈ C ⁢ ( [ 0 , T ] ; L 2 ⁢ ( Ω ) ) ∩ L p ⁢ ( 0 , T ; W 0 1 , p ⁢ ( Ω ) ) of (2.4).

Notice that if r 0 > N , such a solution can be chosen satisfying all the statements of Theorem 3.1.

We observe that the assumption that r 0 ≥ N implies that 2 ⁢ N N + r 0 ≤ p . Hence, if p < 2 (which is not a restrictive condition since we are interested to the limit p → 1 + ), we can apply [33, Theorem 1.5] to the solution u p obtaining that if T ≥ T p ,

(4.12) u p ⁢ ( x , t ) = 0 a.e. in ⁢ Ω × ( T p , T ) ,

where the extinction time T p is defined as follows:

(4.13) T p ≡ | Ω | p ⁢ ( N + r 0 ) - 2 ⁢ N N ⁢ r 0 ( 2 - p ) ⁢ ( r 0 - 1 ) ⁢ ( p r 0 - 2 + p ) p ⁢ C S ⁢ ∥ u 0 ∥ L r 0 ⁢ ( Ω ) 2 - p ,

where C s = ( N - 1 ) ⁢ p N - p . Notice that it results

(4.14) lim p → 1 ⁡ T p = T 1 ≡ | Ω | r 0 - N N ⁢ r 0 ⁢ ∥ u 0 ∥ L r 0 ⁢ ( Ω ) .

Hence, by (4.12)–(4.14) we deduce the assertion (3.4) with T 0 satisfying

T 0 ≤ T 1 .

Notice that if r 0 = N , the value of T 1 is independent of | Ω | since it results T 1 = ∥ u 0 ∥ L N ⁢ ( Ω ) and in this case it results

(4.15) T 0 ≤ ∥ u 0 ∥ L N ⁢ ( Ω ) .

Indeed, being Ω an open bounded set, we know that u 0 ∈ L r 0 ⁢ ( Ω ) with r 0 > N implies u 0 ∈ L N ⁢ ( Ω ) and consequently

u ⁢ ( x , t ) = 0 for all ⁢ t ≥ ∥ u 0 ∥ L N ⁢ ( Ω ) .

Notice that it results

∥ u 0 ∥ L N ⁢ ( Ω ) ≤ ∥ u 0 ∥ L r 0 ⁢ ( Ω ) ⁢ | Ω | 1 N - 1 r 0 = T 1 .

Hence the best estimate here for T 0 when u 0 ∈ L r 0 ⁢ ( Ω ) with r 0 ≥ N is (4.15). Thus the bound (3.5) follows.

To conclude the proof, it remains to show that in the particular case r 0 = N the solution u belongs to L ∞ ⁢ ( Ω × ( ε , + ∞ ) ) for every ε > 0 . To this end, we recall that for every k > 0 arbitrarily fixed, the following estimate on the solution u p holds true (see [33, proof of Theorem 1.5]):

(4.16) | u p ⁢ ( x , t ) | ≤ k for a.e. ⁢ ( x , t ) ∈ Ω × ( T k , p , T )

for every T arbitrarily fixed satisfying T ≥ T k , p , where

T k , p ≡ | Ω | p ⁢ ( N + r 0 ) - 2 ⁢ N N ⁢ r 0 ( 2 - p ) ⁢ ( r 0 - 1 ) ⁢ ( p r 0 - 2 + p ) p ⁢ C s ⁢ ( ∫ Ω ( | u 0 ⁢ ( x ) | - k ) + r 0 ⁢ 𝑑 x ) 2 - p r 0

with C s as above. Hence, recalling the construction of u and observing that

lim p → 1 ⁡ T k , p ≡ T k , 1 ≡ | Ω | r 0 - N N ⁢ r 0 ⁢ ( ∫ Ω ( | u 0 ⁢ ( x ) | - k ) + r 0 ⁢ 𝑑 x ) 1 r 0 ,

we deduce that

| u ⁢ ( x , t ) | ≤ k for a.e. ⁢ ( x , t ) ∈ Ω × ( T k , 1 , T ) ,

and hence, by the arbitrariness of T, we obtain that

| u ⁢ ( x , t ) | ≤ k for a.e. ⁢ ( x , t ) ∈ Ω × ( T k , 1 , + ∞ ) .

Since it results

lim k → + ∞ ⁡ T k , 1 = 0 ,

varying the value of k, we deduce that, for every ε > 0 , u belongs to L ∞ ⁢ ( Ω × ( ε , + ∞ ) ) . ∎

Proof of Theorem 3.3.

Let u be the global solution of (1.1) given by Theorem 2.2. Hence, for every T > 0 arbitrarily fixed, u belongs to C ⁢ ( [ 0 , T ] ; L 2 ⁢ ( Ω ) ) and is the a.e. limit in Ω T of the distributional solutions u p ∈ C ⁢ ( [ 0 , T ] ; L 2 ⁢ ( Ω ) ) ∩ L p ⁢ ( 0 , T ; W 0 1 , p ⁢ ( Ω ) ) of (2.4). We observe that the inequality r 0 < N is equivalent to the bound 1 < 2 ⁢ N N + r 0 . Hence, if u 0 ∈ L r 0 ⁢ ( Ω ) with 1 < r 0 < N (in particular if r 0 = 2 ) applying [33, Theorem 1.2], we know that for every 1 < p < 2 ⁢ N N + r 0 and 1 < r < r 0 it results

(4.17) ∥ u p ⁢ ( t ) ∥ L r ⁢ ( Ω ) ≤ C 1 , p ⁢ ∥ u 0 ∥ L r 0 ⁢ ( Ω ) h 0 , p t h 1 , p ,

where

(4.18) h 0 , p ≡ r 0 ⁢ [ 2 ⁢ N - p ⁢ ( N + r ) ] r ⁢ [ 2 ⁢ N - p ⁢ ( N + r 0 ) ] , h 1 , p ≡ N ⁢ ( r 0 - r ) r ⁢ [ 2 ⁢ N - p ⁢ ( N + r 0 ) ] ,

and

(4.19) C 1 , p = ( 1 ν ⁢ c 2 , p ) 1 ν ⁢ r , ν = γ - 1 , γ = q ⁢ p r ⁢ θ p , q = r - 2 + p p ,

where

θ p = N ⁢ ( r - 2 + p ) ⁢ ( r 0 - r ) r ⁢ [ r 0 ⁢ ( N - p ) - N ⁢ ( r - 2 + p ) ] ,

c 2 , p = r ⁢ ( r - 1 ) ⁢ ( p r - 2 + p ) p ⁢ ( N - 1 ) ⁢ p N - p .

It results

lim p → 1 ⁡ h 0 , p = r 0 ⁢ ( N - r ) r ⁢ ( N - r 0 ) , lim p → 1 ⁡ h 1 , p = N ⁢ ( r 0 - r ) r ⁢ ( N - r 0 ) ,

lim p → 1 ⁡ q = r - 1 , lim p → 1 ⁡ c 2 , p = r ,

lim p → 1 ⁡ θ p = N ⁢ ( r - 1 ) ⁢ ( r 0 - r ) r ⁢ [ r 0 ⁢ ( N - 1 ) - N ⁢ ( r - 1 ) ] , lim p → 1 ⁡ ν = N - r 0 N ⁢ ( r 0 - r ) ,

lim p → 1 ⁡ γ = 1 + N - r 0 N ⁢ ( r 0 - r ) .

Consequently, we have

lim p → 1 ⁡ C 1 , p = C 1 ⁢ ( r 0 ) ≡ ( N ⁢ ( r 0 - r ) ( N - r 0 ) ⁢ r ) N ⁢ ( r 0 - r ) r ⁢ ( N - r 0 ) .

Thus, choosing r 0 = 2 in (4.17) and letting p → 1 + , we deduce for every 1 < r < 2 and t ∈ ( 0 , T ) ,

(4.20) ∥ u ⁢ ( t ) ∥ L r ⁢ ( Ω ) ≤ C 1 ⁢ ∥ u 0 ∥ L 2 ⁢ ( Ω ) 2 ⁢ ( N - r ) r ⁢ ( N - 2 ) t N ⁢ ( 2 - r ) r ⁢ ( N - 2 ) ,

where we have set

(4.21) C 1 = C 1 ⁢ ( 2 ) ≡ ( N ⁢ ( 2 - r ) ( N - 2 ) ⁢ r ) N ⁢ ( 2 - r ) r ⁢ ( N - 2 ) .

Now the result follows thanks to the arbitrariness of T > 0 . ∎

Communicated by Ugo Gianazza

Funding statement: The authors are members of INDAM-GNAMPA and they was supported by it.

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Received: 2024-03-28

Accepted: 2024-08-17

Published Online: 2024-10-02

Published in Print: 2025-07-01

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Decay estimates; singular diffusion equations; 1-Laplacian; Dirichlet problem; total variation flow

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