Local Hadamard well-posedness and blow-up for reaction-diffusion equations with non-linear dynamical boundary conditions

The paper deals with local well-posedness, global existence and blow-up results for reaction--diffusion equations coupled with nonlinear dynamical boundary conditions.

When Q ≡ 0 problem (1) is an initial-boundary value problem related to a semilinear reaction-diffusion equation with homogeneous Dirichlet -Neumann boundary conditions. In this case local well-posedness, under suitable assumptions on f , can be obtained in a standard way using semigroup theory. See for example [42,54] or [2] combined with [14,Appendix]. There is also a wide literature on global existence and blow-up for such type of problems, starting from the classical paper of Levine [32]. See for example [11,22,28,33,34,45], [60,Section 5] and [17,29,46,47,48]. In this case the concavity method of H. Levine is effective in getting blow-up results.
When Q(t, x, u t ) = α(t, x)u t problem (1) consists in a reaction-diffusion equation coupled with a linear dynamical boundary condition. For well-posedness results, obtained by semigroup and interpolation theories we refer to [2,18,19,26,27], while blow-up results were proven in [20,30]. We also refer to [6] for a physical motivation of dynamical boundary conditions, and to the recent papers [21,61,62]. Also in this case the concavity method applies (see [49]) in order to establish blowup. We also would like to mention the classical local-existence and blow-up results in [35,36,38,39] dealing with the related case when the source f appears on the boundary condition.
When Q is nonlinear but monotone increasing in u t and, roughly speaking, either f ≡ 0 or −∆−f is a monotone operator, the existence of global solutions of problem (1) can be proved by applying the results in [16], since the problem can be written as a doubly nonlinear evolution equation in a suitable Banach space. We also refer to [25,43,50] for related results. When Q is nonlinear and f appears on the boundary condition instead than in the equation, local and global existence has been studied in [59]. Next, the same boundary condition arises in the literature in connection with the wave equation, i.e. when the heat operator u t −∆u in (1) is replaced by the wave operator u tt − ∆u. In particular we refer to [4,7,8,12,13,24,31,58]. Finally we would like to mention that our analysis on global behavior of the solutions of (1) is related to the methods in [34]. See also [5,23,37,49].
In this paper we study problem (1) when, roughly, Q(t, x, u t ) ≈ |u t | m−2 u t as |u t | ≥ 1, m > 1, and f (x, u) ≈ |u| p−2 u, p ≥ 2, as |u| ≥ 1. The interest in considering superlinear terms (m > 2) is mainly of theoretical nature. However, a physical model involving Q(t, x, u t ) = u t + |u t | m−2 u t , m > 2, is given in Appendix A.
In order to state and prove our results in the simplest possible way we shall first consider the model problem in Ω where m > 1, p ≥ 2. We denote by 2 * the critical exponent of Sobolev embedding H 1 (Ω) → L q (Ω), i.e. 2 * = 2n/(n − 2) when n ≥ 3 while 2 * = ∞ when n = 1, 2. Moreover we denote · q = · L q (Ω) , · q,Γ1 = · L q (Γ1) for 1 ≤ q ≤ ∞, and the , where u |Γ0 stands for the restriction of the trace of u on ∂Ω to Γ 0 . The first aim of the paper is to show that problem (2) is well-posed in H 1 Γ0 (Ω). The first step in this direction is given by the following result.    . Remark 1. The assumption p ≤ 1 + 2 * /2 in Theorem 1 is quite restrictive when n ≥ 3, although it appears often in the literature quoted above. Clearly it expresses the assumption that the Nemitski operator u → |u| p−2 u is locally Lipschitz from H 1 (Ω) to L 2 (Ω). Such type of assumptions has been overcome, in the author's knowledge, either by getting additional a-priori estimates, as done for example in [7,51], or using linear semigroup and interpolation theories, as done for example in [2,19]. While in this case the nonlinear term Q does not give useful estimates, being active on the boundary, it prevents to use linear theory and interpolation of semigroups. Nonlinear semigroup theory can be used, as in [16], but in this case one still needs to assume that the Nemistski operator above is locally Lipschitz, as in [14]. To prove Theorem 1 we found simpler to first use the monotonicity method of J. L. Lions and then to use a contraction argument. By using the same energy estimates used to prove Theorem 1 we complete our well-posedness analysis as follows.
Theorem 2. (Continuation and local Hadamard well-posedness) Under the assumption of Theorem 1, problem (2) has a unique weak maximal solution u and the following alternative holds: Finally u depends continuously on the initial datum u 0 , that is given any T ∈ (0, T max ) and any sequence (u 0n ) n in H 1 Γ0 (Ω) such that u 0n → u 0 in H 1 Γ0 (Ω), the corresponding weak solution u n is defined in [0, T ]×Ω and u n → u in C([0, T ] ; H 1 Γ0 (Ω)).
1 see Definition 2 below for the precise meaning of weak solutions, which are essentially distributional solutions enjoying a suitable regularity The second aim of the paper is to study the alternative (i)-(ii) in previous Theorem by giving global existence versus blow-up results. When p = 2 it is straightforward to prove that u is global (see Theorem 11 in Appendix B), so we focus on the more interesting case p > 2. Although we are not able to give a complete answer, as usual for nonlinear problems, we give two partial answers when so a Poincarè-type inequality holds (see [63]) and consequently ∇u 2 is an equivalent norm in H 1 Γ0 (Ω). This assumption allows us to use potential-well arguments. In order to state our next results we need to recall the stable and unstable sets introduced in [58]. When p > 2 and (3) holds we introduce the functionals (Ω), and the number When p > 2 and (3), (9) hold true it is easy to see that d > 0. See Lemma 2 below, where two different characterizations of d are given. We define the stable and unstable sets as (Ω) : K(u 0 ) ≤ 0 and J(u 0 ) < d . As an application of Theorem 2 and of a potential-well estimate we give the following global existence result.
Theorem 3. (Global existence) Under the assumptions of Theorem 1 and the further assumptions (9) and p > 2, if u 0 ∈ W s then T max = ∞ and u(t) ∈ W s for all t ≥ 0.
While Theorem 3 can be seen as a simple application of Theorem 2, to recognize that solutions of problem (2) starting in the unstable set blow-up is a more difficult task. When m = 2 this result can be proved by a concavity argument (see [56]), which cannot be applied when m = 2, making this case more interesting. By combining the main technique of [34] with an estimate used in [58] for wave equation we are able to prove the following result.   14) m < m 0 (p) := 2(n + 1)p − 4(n − 1) Remark 2. Even if is not evident from (12) and (13), W s ∩ W u = ∅ (see Lemma 2 below), so Theorems 3 and 4 are consistent. Remark 3. Clearly assumption (14) yields m < p since it is trivial to prove that m 0 (p) ≤ p for p ≥ 2. It strongly reduces the applicability of Theorem 4, as shown by Figure 1 which illustrates the set of the couples (p, m) satisfying (3) and (14). As m 0 (p) > 2 for p > 2, the result is rather sharp in the sublinear case 1 < m ≤ 2, while (3) and (14) force that m < 4 when n = 1, m < 3 when n = 2 and m < 2 + 2 3n−4 when n ≥ 3. This assumption, which looks to be a technical one, comes directly from [58], where it was introduced, and is due to the difficulty in comparing the effect of high order polynomial dissipation, which is related to the L m norm on Γ 1 , with the effect of the source, related to the L p norm on Ω. After nine years from its use, the authors are not aware of any improvement.
As a preliminary step in the proof of Theorem 1 we give a well-posedness result for the problem in Ω where m > 1, T > 0 is arbitrary and g is a given forcing term acting on Ω. Although problem (15) can be studied using the analysis of [16], it is not trivial in that way to get the following result.
and the energy identity holds for 0 ≤ s ≤ t ≤ T . Finally, given any couple of initial data u 01 , u 02 ∈ H 1 Γ0 (Ω) and any couple of forcing terms g 1 , g 2 ∈ L 2 ((0, T ) × Ω), respectively denoting by u 1 and u 2 the solutions of (15) corresponding to u 01 , g 1 and to u 02 , g 2 , the following estimate holds Remark 4. A short comparison with the results which can be obtained by directly applying the abstract results in [16] is in order. Assumptions (A1-2) in [16,Theorem 1] force to restrict to the case m = 2, while the assumption D(B) ⊂ V in [16, implies m ≤ 2(n − 1)/(n − 2) when n ≥ 3. Next one can apply [16,Theorem 4] only when g is more regular in time. Finally, [16,Theorem 5] can be applied only when m = 2.
In order to explain the main difficulties arising in the proofs of our main results we now make some comparison with the arguments used by the second author in [59]. Theorem 5 is essentially proved as [59, Theorem 1.5], even if the necessary adaptations require some care. Theorem 1 is proved by a contraction argument instead that a compactness one. Theorem 2 has no counterpart in [59]. Finally the proof of Theorem 4 requires an untrivial mixing of the technique of [34] with the estimate used in [58], so the authors consider it as the main contribution in the present paper.
The paper is organized as follows. Section 2 deals with some notation and preliminary material, including the proof of Theorem 5, Section 3 is devoted to local well-posedness theory for problem (2) while in Section 4 we study global existence and blow-up for it. Finally the results presented in this introduction are generalized in Section 5 to problem (1), under suitable assumptions on the nonlinearities f and Q. For the sake of simplicity we first present the proofs for the model problem (2) and then we give in (5) the generalizations needed to handle with (1). This section is naturally addressed to a more specialized audience and consequently an higher lever of mathematical expertise of the reader is supposed. In particular most proofs are only sketched.
Moreover we call the trace theorem the existence of the continuous trace mapping H 1 Γ0 (Ω) → L 2 (∂Ω). Moreover the trace of u on Ω will be denoted by u |∂Ω . We also call the Sobolev Embedding Theorem the existence of the continuous embedding H 1 Γ0 (Ω) → L p (Ω) for 2 ≤ p < 2 * . We start by setting the Banach space For elements u ∈ X we shall use the simpler notation u m,Γ1 to mean u |Γ1 m,Γ1 . We now give the precise precise meaning of weak solution of (15).
In order to prove Theorem 5 we need the following Lemma, which extends [53, Theorems 3.1 and 3.2] to the present situation. Its proof consists in a rather technical application of the arguments in [53] which is given in Appendix C for the reader convenience.
in Ω i.e. a function the spatial trace of u on (0, T ) × ∂Ω (which exists by the trace theorem) has a distributional time derivate on (0, T ) × ∂Ω belonging to L m ((0, T ) × ∂Ω), and, for all φ ∈ X and almost all t ∈ [0, T ] the function u satisfies Then (Ω)) and the energy identity Proof of Theorem 5. To prove the existence of a weak solution of (15) we apply the Faedo-Galerkin procedure. Let (w k ) k be a sequence of linearly independent vectors in the space X, which was defined in (20), whose finite linear combinations are dense in it. By using the Graham-Schmidt orthonormalization process, we can take (w k ) k to be orthonormal in For any fixed k ∈ N we look for approximate solutions of (15), that is for solutions In order to recognize that (30) has a local solution, we set and H k (t) = Ω g(t, x)B k (x)dx, so problem (30) can be rewritten as Then, using the arguments in [59, Proof of Theorem 1.5] we get that G k is an homeomorphism from R k into iteself, with inverse G −1 k , and that (34) has a solution y k ∈ W 1,1 (0, t k ) for some t k ∈ (0, T ], and consequently (30) has a solution u k ∈ W 1,1 (0, t k ; X). Moreover, since G k (y)y ≥ |y| 2 for all y ∈ R k , by the Schwartz inequality it follows that |y| ≤ |G k (y)|. Then G −1 k (y) ≤ |y| for all y ∈ R k , so that Multiplying (30) by (y j k ) and summing for j = 1, . . . , k, we obtain the energy identity (here and in the sequel, explicit dependence on t will be omitted, when clear) (36) d dt Integrating over (0, t), 0 < t < t k , and using Young inequality, we get Then, using (29), there exists C = C ∇u 0 2 , g L 2 ((0,T )×Ω) > 0 such that (37) and Hölder inequality in time it follows that (38) yields that |y k (t)| ≤ C . Then, by (35) . We can then apply [15, Theorem 1.3, Chapter 2] to conclude that t k = T for k = 1, . . . , n. Next, by (37) and (38), it follows that, up to a subsequence, . A consequence of the convergences (39) and of Aubin-Lions compactness Lemma (see [10,3,52]) is that u k → u strongly in C([0, T ] ; L 2 (Ω)), so that u(0) = u 0 . It follows in a standard way (see, for example, [57, p. 272 Next, multiplying (30) by φ ∈ C ∞ c (0, T ), integrating on (0, T ), passing to the limit as k → ∞ (using (39)) and finally using the density of the finite linear combinations of (w k ) k in X, we obtain T 0 . Consequently (u t , w) + (∇u, ∇w) + Γ1 χw = Ω gw almost everywhere in (0, T ). Then to prove that u is a weak solution of (15) we have only to show that By Lemma 1 we obtain (27) and the energy identity The classical monotonicity method (see [41] or [59, p. 186]) then allows us to prove (40).
Finally, to prove the estimate (19), which also yields the uniqueness of the solution, Then, by Lemma 1, using the monotonicity of the map x → |x| m−2 x we get the estimate By Young inequality . By combining the last two estimates we get (19) and conclude the proof.

Proofs of Theorems 1 and 2.
This section is devoted to prove our main well-posedness Theorems 1 and 2. We first precise the meaning of weak solution for problem (2).
(Ω). When assumption (3) holds we say that u is a weak solution of problem (2) in [0, T ] × Ω if (a-d) of Definition 1 hold, with the distribution identity (21) being replaced by Moreover we say that u is a weak solution of problem (2) Remark 5. Since p ≤ 1 + 2 * /2 and ϕ ∈ H 1 Γ0 (Ω) the integral in the right-hand side of (42) makes sense due to the Sobolev Embedding Theorem.
Proof of Theorem 1. We set, for any 0 < T < ∞, the Banach space (Ω)) , and the closed convex set X T = {u ∈ Y T : u(0) = u 0 }. Let u ∈ X T . By (3) we have 2(p − 1) ≤ 2 * and then, by the Sobolev Embedding Theorem, for some K 0 = K 0 (Ω) > 0 (in the sequel of the proof K i , i ∈ N, will denote suitable positive constants depending on p, n and Ω). Hence |u| p−2 u ∈ L ∞ ((0, T ); L 2 (Ω)). Then by Theorem 5 there is a unique weak solution v of the problem in Ω. Moreover × Ω) and the energy identity where v denotes the solution of (44) that corresponds to u. We are going to prove that we can apply the Banach Contraction Theorem to Φ : We first claim that Φ maps B R into itself for R sufficiently large and T small enough. Let u ∈ B R . By (45) and (43) we get, for t ∈ [0, T ], Now using Young inequality it follows that, for all t ∈ [0, T ], Consequently, by (46), Using Hölder inequality we have v(t) 2 and so, by (49), Now restricting to T ≤ 1 we have T 2 ≤ T and so combining (48) and (50) we get By (51) in order to prove that v ∈ B R , it is enough to show that 5R 2 0 ≤ 1 2 R 2 and 6K 2 R 2(p−1) T ≤ 1 2 R 2 . Hence our claim holds for In the sequel we shall assume that (52) holds.
In order to prove that the solution is unique we use a standard procedure of ODEs, using previous claims, which is briefly outlined as follows. Let u, u be two weak solutions of (2)  Proof of Theorem 2. The existence of the unique maximal solution u of (2) follows by Theorem 1 in a standard way: first one sets U to be the set of all tweak solutions of (2), then one proves that any two elements of U must coincide on the intersection of their domains, arguing as at the end of previous proof, finally one defines u(t) to coincide with any of these solution for t in the union of the domains.
Next, in order to prove that the alternative (i)-(ii) holds, let us suppose, by contradiction, that Then there is a sequence T n → T − max such that u(T n ) H 1 , m, p, Ω, Γ 1 ) is independent on n. This leads to a contradiction, since, in this way, we can continue the solution to the right of T max . Now, in order to prove that u depends continuously on the initial datum, we fix T ∈ (0, T max ) and we denote M = u C([0,T ];H 1 Γ 0 (Ω)) . Since u 0n → u 0 in H 1 Γ0 (Ω) there is n 1 ∈ N such that u 0n H 1 for all n ∈ N. Now we define w n = u n − u, which is a weak solution of the problem in Ω in the sense of Lemma 1. Consequently Then, keeping the notation of the proof of Theorem 1 and using the arguments already used to prove (58) together with (67) we get the estimate for any ε > 0. Consequently, for ε > 0 sufficiently small we have where C 3 = C 3 (p, n, Ω, u 0 , T ) > 0. Moreover, since T * ≤ 1, by using Hölder and so by (70) Combining (70) and (71) we get follows. In particular we have Then, since u 0n → u 0 as n → ∞, for n ≥ n 2 , with n 2 sufficiently large, we have u n (T * ) H 1
Proof. An easy calculation shows that for any u ∈ H 1 Γ0 (Ω)\{0} we have max Hence, by (75), In order to show that W s = W 1 = W 1 we first prove that W s ⊆ W 1 . Let u 0 ∈ W s and suppose, by contradiction, that ∇u 0 ≥ λ 1 . Since J(u 0 ) < d = E 1 and u 0 p p ≤ ∇u 0 2 2 it follows that which contradicts (76). By (75), since λ 1 = B 1 λ 1 , one immediately gets that W 1 ⊆ W 1 . To prove that W 1 ⊆ W s , let u 0 ∈ W 1 . By (75), (78) and (76)  and so K(u 0 ) ≥ 0. In order to show that W u = W 2 = W 2 we first prove that W 2 ⊆ W u . Let u 0 ∈ W 2 and suppose, by contradiction, that K(u 0 ) > 0. So u 0 p p < ∇u 0 2 2 by (10). Moreover, J(u 0 ) < d = E 1 and ∇u 0 2 > λ 1 . Then it follows that which contradicts (76). By (75) one immediately gets that W 2 ⊆ W 2 . To prove that W u ⊆ W 2 and conclude the proof, we take u 0 ∈ W u . We note that, by (75), we have In what follows we shall use the following derivation formula, which is proved here for the sake of completeness only.
We now show that W s and W u are invariant under the flow generated by (2).

Lemma 4.
Under the assumptions of Theorem 1, let u be the weak maximal solution of problem (2). Also assume that (9) holds. Then Proof. By Lemma 3, the energy identity (6) can be written as Consequently t → J(u(t)) is decreasing in [0, T max ) and by Lemma 2 On the other hand, by (75) we have the inequality J(u(t)) ≥ g ( ∇u(t) 2 ), where g(λ) = λ 2 /2−B p 1 λ p /p for λ ≥ 0. It is straightforward to verify that g is increasing in [0, λ 1 ) and decreasing in [λ 1 , ∞), so λ 1 is the maximum point for g, and that g(λ 1 ) = E 1 . Consequently, by (83) we have ∇u(t) 2 = λ 1 for all t ∈ [0, T max ). Since the function t → ∇u(t) 2 is continuous, by Lemma 2 the proof is complete.
Proof of Theorem 3. By Theorem 2 we just have to prove that when u 0 ∈ W s the alternative (8) in Theorem 2 leads to a contradiction, which is obtained by combining Lemma 4-(i) with the Poincarè type inequality recalled at the beginning of the section.
In this way (here and in the sequel of the proof explicit dependence on t will be omitted) we obtain the identity We estimate the two terms in right-hand side of (88) separately. By Hölder inequality we get To estimate the L m (Γ 1 ) norm of u |Γ1 we first recall the trace embedding for Sobolev space of fractional order (see [1,Theorem 7.58,p. 218] and [55]) H s (R n ) → W χ,l (R n−1 ) when 2 ≤ l < ∞, χ = s− n 2 + n−1 l > 0. Since W χ,l (R n−1 ) → L l (R n−1 ), using the C 1 regularity of Ω and a standard partition of the unity we have the trace embedding H s (Ω) → L l (∂Ω) when 2 ≤ l < ∞, s − n 2 + n−1 l > 0 and 0 < s ≤ 1. Using the last embedding with l = max{2, m}, the fact that ∂Ω has finite surface measure and Hölder inequality we get Next, by the interpolation inequality (see [40, p.49] 3 ) and the already quoted Poincarè type inequality, we have Actually interpolation inequality is stated in the quoted reference only for C ∞ domains Ω, but as explicitly remarked there this assumption is not optimal. In particular, since 0 < s ≤ 1, the C 1 regularity assumed here is sufficient to prove the result. Unfortunately the authors were not able to find a reference where interpolation inequality is stated under optimal regularity assumptions.

More general results
This section is devoted to generalize our results to problem (1), where Q and f satisfy suitable assumptions which generalize the specific behaviour of |u t | m−2 u t and |u| p−2 u. Our assumptions on Q are the following ones.
Remark 6. When Q = Q(v) assumptions (Q1)-(Q2) reduce (independently on Θ) to assume that Q ∈ C(R) is increasing and such that ). Remark 7. Let us note, for a future use, that (Q1)-(Q2) yield the existence of positive constants c 5 and c 6 (possibly dependent on Θ) such that and for almost all (t, x) ∈ (0, Θ) × Γ 1 and all v ∈ R.

Forced heat equation.
We first present our generalization of Theorem 5 to the problem (109) where g is a given term acting on Ω and T > 0 is fixed.
We set Φ : X T → X T by Φ(u) = v. By using the same arguments in the proof of Theorem 1 together with assumptions (Q2) and (F1) we get for any u ∈ B R , the estimate which generalizes (47) to this more general situation, where now the constants K i depends also on c 7 . Then we proceed as in the quoted proof with (R 2 + R 2(p−1) ) replacing R 2(p−1) . Consequently we get that Φ(B R ) ⊂ B R provided that (119) R = 4R 0 , and T ≤ min 1, Θ, K 3 (16 + 16 p−1 R 0 2(p−2) ) −1 , generalizing (52). In order to show that, for suitable T , Φ is a contraction in B R we proceed exactly as in the quoted proof by taking u, Clearly, w is a weak solution of the problem (120) generalizing (53). Since by (114) we have f (·, u), f (·,ū) ∈ L ∞ (0, T ; L 2 (Ω)) and by (107) we have Q(·, ·, v t ), Q(·, ·,v t ) ∈ L m ((0, T ) × Γ 1 ) we can apply Lemma 1 to get Using (F1) and (Q2) we generalize the estimate (56) to the following one Consequently exactly the same arguments used in the quoted proof allow to prove the estimate (Ω)) .
replacing (63), so by (119) 1 , Φ is a contraction provided T < K −2 We can the finally fix T * and complete the proof.
The following result is nothing but the generalization of Theorem 2.
Sketch of the proof. We describe the adaptations needed to cover this more general situation with respect to the arguments used in the proof of Theorem 2. The existence of a unique weak maximal solution of (1) follows exactly in the same way. When proving the alternative (i-ii), since the equation is not autonomous (as the term Q is explicitely time-dependent) a more detailed explanation is needed. Let us suppose by contradiction that (66) holds, so there is a sequence T n → T − max < ∞ such that u(T n ) H 1 Γ 0 is bounded. Since Q satisfies assumptions (Q1-2) for all positive Θ, we can choose Θ = T max + 1. We set for any n ∈ N the time-shifted nonlinear term Q n (t, x, v) = Q(t + T n , x, v), which satisfies assumptions (Q1-2) with Θ = Θ n := T max − T n + 1 ≥ 1, so that Q n satisfies the same assumptions for Θ = 1 for all n ∈ N. It follows that the existence time T * assured by Theorem 7 is independent on n, so problem (1) with initial time T n and initial datum u(T n ) has a unique weak solution in [T n , T n + T * ] × Ω, which leads to the desired contradiction.
When proving the continuous dependence of the solution u on the initial datum we get the energy identity  (3) we then get the estimate (69) again, so we can conclude the proof exactly as in Theorem 2.

5.3.
Global existence versus blow-up. In order to generalize Theorems 3 and 4 to problem (1) we first generalize Lemma 3. We introduce the notation for almost all t ∈ (0, T ).
Proof. We first note that an immediate consequence of (114) is that |F (x, u)| ≤ c 9 (1 + |u| p ) for a positive constant c 9 . Hence Ω F (·, u) ∈ L ∞ (0, T ) ⊂ L 2 (0, T ). Consequently exactly the same arguments used in the proof of Lemma 3 apply to this more general case.
To extend in a suitable way the definition of the stable and unstable sets we need to introduce a second structural assumption on the nonlinearity f .
(F2) There is c 10 ≥ 0 such that F (x, u) ≤ c 10 p |u| p for almost all x ∈ Ω and all u ∈ R.
Proof. By Lemma 5 the energy identity (116) can be written as By (129) and (Q2) the energy function E(t) := J(u(t)) is decreasing in [0, T max ). Hence (83) continue to hold. By (124) we have J(u(t)) ≥ g( ∇u(t) 2 ), where Then, when D 1 > 0, the same arguments used in the proof of Lemma 4 apply, while there is nothing to prove when D 1 ≤ 0.
We can now state the generalization of Theorem 3.
Theorem 9. Under the assumptions of Lemma 6 if u 0 ∈ W s then T max = ∞ and u(t) ∈ W s for all t ≥ 0.
Proof. When D 1 > 0 we can exactly repeat the proof of Theorem 3 by using Lemma 6. When D 1 ≤ 0 the same argument applies since in this case we have J(u) ≥ 1 2 ∇u 2 2 so W s is bounded.
In order to generalize Theorem 4 we need to strengthen assumption (Q1-2) to the following ones.
Remark 9. We remark that the nonlinearities Q 0 and Q 1 defined in Remark 6 satisfy as well assumption (Q1 -2 ). Moreover when Q = Q(v) these assumptions reduce to assume that Q ∈ C(R) is increasing and We also note, for future use, some further consequences of (Q1 ) and (Q2 ). Since for almost all (t, x) ∈ (0, ∞) × Γ 1 , where c 5 and c 6 are positive constants.
In order to state our blow-up result for problem (1) we need a further specific structural assumption on f .
Clearly the model nonlinearity f 0 defined in (112)  Remark 10. Assumption (132) needs some comment, as it express the possible time-behavior of Q. When d(t) = (1 + t) β , β ∈ R, it reduces to β ≤ µ − 1, and in particular when µ = m in assumption (Q1 ) (what happens for example when Q(v) = d(t)|v| m−2 v), it reduces to β ≤ m − 1, which is a well-known optimal assumption to prevent over-damping for time dependent damping terms in ordinary differential systems.
Proof. As in the proof of Theorem 4 we prove, by contradiction, that there are no solutions in the whole (0, ∞) × Ω. We fix E 2 ∈ (J(u 0 ), E 1 ) and set H by (84). By using Lemma 6 and (124) we get a slightly generalized version of (85), that is By (Q2 ) formula (86) is now generalized to so that (87) holds true. By (107) we can again take φ = u t in the distribution identity (115) so getting the following generalized version of (88) The estimate (98) of the second term in the right hand side of (135) keeps unchanged, while the estimate the first term in it needs a more detailed explanation. We use (130), (131) and Hölder inequality twice to get by j(t, x) = − ∂u ∂ν , since ρ = 1. Finally, the thermal contact of the fluid at Γ 1 yields the continuity condition u(t, x) = v(t), x ∈ Γ 1 , t ≥ 0, while the temperature on Γ 0 is assumed to be constant (for simplicity constantly vanishing), that is Combining (138)-(139), we obtain (1) with f = |u| p−2 u and Q = u t + |u t | m−2 u t . These nonlinear terms are included in theory developed in Section 5. In particular Theorem 10 shows that the refrigerating system cannot avoid the internal explosion with this conditions.  By Gronwall inequality we then get that t 0 u t 2 2 is bounded up to T max . By (141) we consequently get that also u 2 is bounded. Hence, by (140) also ∇u 2 is bounded. So we contradict (8) and conclude the proof. The key point is to show that the energy identity holds. With this aim and fixed 0 ≤ s ≤ t ≤ T , we set θ 0 to be the characteristic function of the interval [s, t].