Well-posedness for a class of nonlinear degenerate parabolic equations

In this paper we obtain well-posedness for a class of semilinear weakly degenerate reaction-diffusion systems with Robin boundary conditions. This result is obtained through a Gagliardo-Nirenberg interpolation inequality and some embedding results for weighted Sobolev spaces.

1.1. Problem formulation. We consider the problem (1) under the following assumptions: x, u) is loc. absolutely continuous for a.e. x ∈ (−1, 1), ∀u ∈ R, • there exist constants γ 0 ≥ 0, ϑ ∈ [1, ϑ sup ) and ν ≥ 0 such that Remark 1. The following is an example of function f that satisfies the assump-  Remark 2. We note that Theorem 1.1 holds also for the SDP (1) under the assumptions (A.1)−(A.4) and (A.5 SD ), with weighted Neumann boundary conditions. This result has already been obtained by the author in [7].
1.3. Structure of this paper. The main theorem, Theorem 1.1, is proved in Section 3, with general Robin boundary conditions, applying the new Gagliado-Nirenberg interpolation inequalities and the embedding results for weighted Sobolev spaces obtained in Section 2 (references to "interpolation inequalities" can be found in [2], [6], [10] and [11]). In Section 4, we show an application to the global approximate multiplicative controllability for system (1) and we present some perspectives for this kind of controllability.
2. Interpolation inequalities and embedding results for weighted Sobolev spaces. In this section, first we introduce some weighted Sobolev spaces and we obtain a Gagliardo-Nirenberg interpolation inequality, then we prove embedding results for spaces involving time.

2.2.
The operator (A, D(A)). In this work we consider the operator (A, D(A)) defined by where α ∈ L ∞ (−1, 1). In [4] we showed that A is a closed, self-adjoint, dissipative operator with dense domain in L 2 (−1, 1). Therefore, A is the infinitesimal generator of a C 0 − semigroup of contractions in L 2 (−1, 1).
moreover, for every q ≥ 1 2 there exists a positive constant c, c = c(q), such that where α = 2 2+q .

D(A)
is the domain of the operator defined in (5). 5 It is well known that this norm is equivalent to the Hilbert norm Moreover, for every p ∈ (2, 5), owing to (11) and applying Hölder's inequality (with conjugate exponents: 3 p−2 and 3 5−p ) we obtain The proof of the following Lemma 2.5 has not been included in this paper as is similar to that of Lemma 3.5 of [7].
3. Existence and uniqueness of solutions of semilinear WDP (1). In this section, in order to study the semilinear W DP (1), we represent it in the Hilbert space L 2 (−1, 1) as where A is the operator defined in (5), u 0 ∈ L 2 (−1, 1), and, for every u ∈ B(Q T ), the Nemytskii operator associated with the W DP (1) is defined as Now, we will deduce the following proposition.
3.1. Strict solutions of the W DP (1). In order to continue, the next definition is necessary.
Definition 3.1. If u 0 ∈ H 1 a (−1, 1), u is a strict solution of the W DP (1), if u ∈ H(Q T ) and a.e. in Q T := (0, T ) × (−1, 1) Now, we give the following existence and uniqueness result for strict solutions. The proof of Theorem 3.2 has not been included in this paper as it is very lengthy and technical. However, this proof is similar to that of the existence and uniqueness of strict solutions for the SDP (1), proved in the Theorem 3.15 of [7] (see also Appendix B of [7], and the author's Ph.D. Thesis [8]). The fundamental idea of this proof uses the "fixed point" argument. The only difference between the SD case and the WD case consists in the fact that, in the WD case, there is the need to adapt the Robin boundary condition, but the sign condition on the coefficients (see (A.5 W D )) assists us.

3.2.
Strong solutions of the W DP (1). The following notion of "strong solutions" is classical in PDE theory, see, for instance, [1], pp. 62-64 (see also [7]). Definition 3.3. Let u 0 ∈ L 2 (−1, 1). We say that u ∈ B(Q T ) is a strong solution of the WDP (1), if u(0, ·) = u 0 and there exists a sequence {u k } k∈N in H(Q T ) such that, as k → ∞, u k −→ u in B(Q T ) and, for every k ∈ N, u k is the strict solution of the Cauchy problem with initial datum u k (0, x).

Remark 4.
We note that, due to the definition of the B(Q T )−norm (see Section 3.1), due to the fact that, as k → ∞, u k −→ u in B(Q T ), from the Definition 3.3 we deduce that u k (0, ·) −→ u 0 in L 2 (−1, 1). Moreover, since φ is locally Lipschitz continuous (see Proposition 1), Now, we give the following proposition. 1)). u, v are strict (or strong) solutions of the W DP (1), with initial data u 0 , v 0 respectively. Then, we have where C T = e (ν+ α + ∞ )T and α + denotes the positive part of α ( 7 ).
The proof of Proposition 2 is similar to that of Proposition 3.16 of [7].

3.3.
Proof of the main result. Finally, we can prove the main result of this paper, that is, the existence and uniqueness of strong solutions to W DP (1) with initial data in L 2 (−1, 1).
Proof. (of Theorem 1.1). Let u 0 ∈ L 2 (−1, 1). There exists {u 0 k } k∈N ⊆ H 1 a (−1, 1) such that, as k → ∞, u 0 k → u 0 in L 2 (−1, 1). For every k ∈ N, we consider the problem (15) with initial datum u k (0, x) = u 0 k (x), x ∈ (−1, 1). For every k ∈ N, through the uniqueness and existence of the strict solution to system (15) (see Theorem 3.2), there exists a unique u k ∈ H(Q T ) strict solution to (15). Then, we consider the sequence {u k } k∈N ⊆ H(Q T ) and by direct application of the Proposition 2 we prove that {u k } k∈N is a Cauchy sequence in the Banach space B(Q T ). Then, there exists u ∈ B(Q T ) such that, as k → ∞, u k → u in B(Q T ) and u(0, ·) 4. Some applications and prospectives: global multiplicative controllability. We are interested in studying the nonnegative multiplicative controllability of (1) using the bilinear control α(t, x). Some references for multiplicative controllability are [7], [3], [4] and [5]. Let us start with the following definition.

4.1.
Multiplicative controllability for SDPs. In the following theorem, proved in [7], the nonnegative global approximate controllability result is obtained, for the semilinear SDP (1)

4.2.
Multiplicative controllability for WDPs. Thanks to the well-posedness result for WDPs obtained in this paper, we will be able to investigate the possibility of extending Theorem 4.2 from SDPs to WDPs (see [9]).