Renormalized solutions to a reaction-diffusion system applied to image denoising

This paper concerns the Neumann problem of a reaction-diffusion system, 
which has a variable exponent Laplacian term and could be applied to image denoising. 
It is shown that the problem admits a unique renormalized solution for each integrable 
initial datum.

It is clear that g, p ∈ C ∞ (Ω) and g 1 ≤ g(x) ≤ g 2 , 1 < p(x) < 2, x ∈Ω with g 1 , g 2 > 0. The system (1)-(4) is used to restore the noisy image, where u represents the restoration image describing a real scene (the unknown), f is the observed noisy image and w represents the oscillatory of u. Image restoration is a fundamental problem in both image processing and computer vision with numerous applications. In the last twenty years, nonlinear partial differential equations have become a major method for image restoration. Y. Chen, S. Levine and M. Rao [11] firstly proposed a partial differential equation with variable exponent for image restoration and established the well-posedness of weak solutions. Later, many authors proposed similar equations to refine the models ( [1,17,18]). Particularly, in [17] we consider the following model for image restoration, which combined the H −1 norm for oscillatory functions with the variable exponent regularization where f ∈ L 2 (Ω). Minimizing E H −1 and using gradient descent algorithm, we could formally deduce the Euler-Lagrange equations (1) and (2). In [17] we proved the well-posedness of weak solutions and showed the effectiveness of the model in image restoration by numerical experimental. While in this present paper, we treat the general case f ∈ L 1 (Ω), i.e., the noisy data is allowed to be an integrable function. The motivation to treat this case is natural since images may be not smooth enough ( [16]) and it is hoped to recover the image in more extensive space. In the mathematical study, for the case f ∈ L 2 (Ω), one can establish wellposedness of weak solutions to the problem (1)-(4) in a suitable Sobolev space, which satisfy the equations in the distribution sense. However, there may not exist such a weak solution for an integrable datum. Therefore, a different solution should be defined for the problem (1)-(4) with f ∈ L 1 (Ω). In this paper, we select its renormalized solutions, which were firstly introduced by Di Perna and Lions [13] for the study of Boltzmann equation and then are used to study parabolic equations ( [2], [8], [9], [14], [24], [26]) and even parabolic equations with variable exponent Laplacian term ( [6,29]). In practice, in order to avoid nonsensical results such as negative gray levels, ones need to cut off gray levels to reduce the loss of dynamic range in practical computation ( [15]). So, renormalized solutions are also more suitable than weak solutions. Compared with previous results, the contribution of our work is to study a reaction-diffusion system with variable exponent Laplacian term and Neumann boundary conditions. So the methods used in [6] and [29] are not suited entirely for our problem. This is worthy mentioning that the similar problem for the case of p(x) ≡ p and Neumann boundary conditions is well studied in [2,3,4,5]. Finally, we consider the regularity with the initial data in L r (Ω) for 1 ≤ r < 2, which has been discussed in [25] and [27] for p-Laplican equation and in [6] for p(x)-Laplacian equation.
The paper is arranged as follows. In Section 2, we first recall some mathematical preliminaries about variable exponent spaces, and then state the definition of the renormalized solutions and the main results. Section 3 is devoted to the proof of the main results.
2. Mathematical preliminaries and main results. In this section, we first recall some results on generalized Lebesgue-Sobolev spaces L p(x) (Ω) and W 1,p(x) (Ω) which could be found in [12] and then state the definition of renormalized solutions and the main results on the well-posedness. We always assume p(x) is a continuous function onΩ satisfying p − = min which is equipped with the Luxemburg norm The space L p(x) (Ω) is also called a generalized Lebesgue space. Next, define the variable exponent Sobolev space W 1,p(x) (Ω) by which is a Banach space equipped with the norm The function spaces L p(x) (Ω) and W 1,p(x) (Ω) have the following properties: • Both L p(x) (Ω) and W 1,p(x) (Ω) are separable and reflexive Banach spaces.

2.2.
The definition of renormalized solutions. Recall the following space introduced in [3,23]. Denote T K the truncation function at height K ≥ 0: and define the space Similar to the result of Lemma 2.1 in [7] and Proposition 4.1 in [23], we have the following lemma. Note that this definition of derivative is not a definition in the sense of distributions.
Now, we give the definition of renormalized solutions of the problem (1)-(4).
Next, we study the convergence of {u n } and {w n } in C([0, T ]; L 1 (Ω)). Proof. Let n and m be two positive integers. It follows from (14) and (15) that and t 0 Ω The Gronwall inequality shows Letting n, m → ∞ and then δ → 0 + in (24), one obtains By lemma 3.1 and lemma 3.3, we conclude that there exist two subsequences of {u n } and {w n }, still denoted by themselves for convenience, such that u n converges to a function u in C([0, T ]; L 1 (Ω)), w n converges to a function w in C([0, T ]; L 1 (Ω)) and To study the strong convergence of ∇T K (u n ) and ∇T K (w n ), we first list some results about the time regularization of T K (u) (for fixed K > 0), which have been exploited in [20], [9], [6], [29], etc. For any fixed K > 0 and µ > 0, as the method of [6], we take a sequence of functions defined in Ω such that v µ by extending T K (u) by 0 for s < 0. Then T K (u) µ is a unique solution of the monotone problem: in Ω.
with T K (u) µ ∈ L ∞ (Q T ) and ∇(T K (u)) µ ∈ (L p(x) (Q T )) N . It is also clear that (T K (u)) µ → T K (u) a.e. in Q T and weakly- * in L ∞ (Q T ) as µ → ∞, The proof of the following lemma is firstly proved in [9] with constant exponents and Dirichlet boundary cases. To make this paper self-contained, we include a proof in the variable exponents and Neumann boundary cases with the similar method.
Proof. Since S is increasing and S(r) = r for |r| ≤ K, T K (S(u n )) = T K (u n ), and T k (S(u)) = T K (u) a.e. in Q T .
As a consequence (T K (S(u))) µ = (T K (u)) µ , a.e. in Q T , for any µ > 0. It follows that under the notation v n = S(u n ), and v = S(u), we have Since S is bounded, we have v n → v strongly in L 2 (Q) and in L ∞ weak-* as n → ∞.

Then lim
Since T K (S(u 0 )) = T K (u 0 ) a.e. in Ω, we have and By (32)-(34), we have and then lim inf The dominated convergence theorem gives g(x)|∇u n | p(x)−2 ∇u n S m (u n )∇(T K (u)) µ dxdτ dt. Noting a.e. in Q T , one obtains as n → ∞. Owing to S m (r) = 1 for |r| ≤ m, one gets that for any m ≥ K, Note that for m ≥ K, a.e. (x, t) ∈ Q T ∩ {|u n | < K}.
Thus for m ≥ K, Combining (39)-(41) leads to lim sup Similarly to Lemma 5 in [10], one could get that ∇T K (u n ) converges to ∇T K (u) a.e. in (Q T ) and then in (L p(x) (Q T )) N for any T < T . By extending the functions to a large interval (see [9]), it is then sufficient to prove that ∇T K (u n ) converges to Turn to the convergence of ∇T K (w n ). Denote V n µ = T K (w n ) − (T K (w)) µ and choose ψ = S m (w n )V n µ in (15). One can get that Similarly, one gets that ∇T K (w n ) converges to ∇T K (w) in (L 2 (Q T )) N .
Proof of the existence. Let us verify that (u, w) satisfies the definition of the weak solution. Owing to Lemma 3.3, (25)- (27), (5) and (6) hold. For fixed positive integer m, Letting n → ∞ and then m → ∞, it follows from Lemma 3.2 that Similarly, one can prove lim m→∞ {m≤|w|≤m+1} For S ∈ C ∞ (R) with S ∈ C 0 (R), multiplying (10) by S (u n ) and (11) by S (w n ), one gets and Due to Lemma 3.3, as n → ∞, we have (S(u n ), S(w n )) converges to (S(u), S(w)) in C([0, T ]; Ω) and weakly− * in L ∞ (Q T ) and and and S (w n )∇w n = S (w n )∇T K (w n ) a.e. in Q T .
Owing to Lemma 3.3 and Lemma 3.5, one gets and S (w n )∇w n S (w)∇w weakly in L 2 (Q T ).
It follows from Lemma 3.5 that and and Using (44)-(51), one gets (8) and (9) by letting n → ∞ in (42) and (43). Now, we shall give the proof of the uniqueness result of Theorem 2.3. We first give the following integration by parts lemma, which is slight modification of Lemma 4.1 in [5]. Lemma 3.6. Let (u, w) be the renormalized solution of the problem (1)-(4). Then for any continuously differentiable nondecreasing functions G(r) and H(r), we have Based on the method in [9], for s > 0 and σ > 0, define S σ s ∈ W 2,∞ (R) by It is obvious that Then the following technical lemma for renormalized solutions can be proved.
Turn to the regularity of the renormalized solution to the problem (1)-(4) under the assumption that f ∈ L r (Ω), 1 ≤ r < 2 and p − > 2 − 1/(N + 1). Since we prove the existence of the renormalized solutions by an approximation technique, we can assume additionally that the solution is smooth enough. So by the standard theory of parabolic equations [21], one has Lemma 3.8. If u ∈ L r (Q T ), f ∈ L r (Ω) and w is the solution to (2), then Furthermore, we prove Lemma 3.9. If f ∈ L r (Ω) and u is the solution to (1), then Proof. Taking ϕ = (|u| + 1) r−1 sign(u) in (1) yields Integrating over (0, t) and Using the Hölder inequality, one gets Then, the lemma is proved by the Growall inequality.

QIANG LIU, ZHICHANG GUO AND CHUNPENG WANG
Define p − i and p + i to be the minimum and the maximum of p on B i , respectively. Moreover, α − i , α + i , β − i and β + i are defined similarly. Then Similarly, Let p * i = N p − i /(N − p − i ) and σ i = p − i (1 + r/N ). For sufficiently large K > 0, it follows from the interpolation in Lebesgue spaces, the Poincaré inequality, Lemma 3.9 and Lemma 3.1 that where The Tchebycheff inequality gives meas{|u| > K, x ∈ Q i } ≤ CK 2−r−σi .