Global small data solutions for semilinear waves with two dissipative terms

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Introduction
In this paper, we study the following Cauchy problem for semilinear wave equations with two dissipative terms: with 0 ⩽ < 1∕2 < ⩽ 1 , where the nonlinearity F = F(u, u t ) on the right-hand side can be described by (1) x ∈ ℝ n , 1 3 The critical exponent for the Cauchy problem (1) with F = |u| p or F = |u t | p , is, respectively, given by p 0 (n, ) or p 1 (n, ) , where If p ∈ (1, p j (n, )] (subcritical and critical cases), it is easy to prove that problem (1) admits no global (in time) weak solution, for initial data verifying a suitable sign assumption (Proposition 2.1). If p > p j (n, ) (supercritical case), then we prove that there exists a unique global (in time) solution for sufficiently small data (Theorem 2.1 and Corollary 2.1) in an appropriate space. In Theorems 2.1 and 2.2 we prove the existence of global (in time) energy solutions u ∈ C [0, ∞), L ∩ H 2 ∩ C 1 [0, ∞), L 2 to (1) with F = |u| p where, respectively, = 1 or ∈ (1,2] , assuming small initial data in L ∩ L 2 . We call them "energy solutions" since the energy functional is well-defined and continuous. In Theorem 2.3, we do not consider energy solutions, but we prove the existence of global (in time) weak solutions u ∈ C 1 ([0, ∞), L ∩ L p ) to (1) with F = |u t | p , for some > 1 , assuming small initial data in L ∩ L p . In the last years, dissipative wave equations attracted a lot of attention. Let us first consider the linear wave equation with friction and viscoelastic damping, namely and the corresponding nonlinear problem. Asymptotic profiles of solutions to (5) are derived in [1,2]. In [3], the second author showed that the presence of two damping terms in (5) allows to derive L p − L q estimates in the full range 1 ⩽ p ⩽ q ⩽ ∞ , in any space dimension n ⩾ 1 . These estimates may be effectively used to study global (in time) existence of small data solutions to where F = F(u, u t ) is the same as in (2). In particular, these estimates allow us to prove global (in time) existence of small data solutions for any p > 1 in the case F = |u t | p . Other studies on doubly dissipative wave models can be found in [4,5] and in the references therein.
The main advantage of the presence of two dissipative term in (5) may be understood noticing that the value = 1∕2 is a threshold between two different asymptotic profiles for the solution to the damped wave equation (2) F(u, u t ) = | j t u| p with p > 1, and j = 0 or j = 1.
x ∈ ℝ n , t > 0, u(0, x) = u 0 (x), u t (0, x) = u 1 (x), x ∈ ℝ n , According to the classification introduced in [6], the case ∈ [0, 1∕2) corresponds to an effective dissipation. In this case, the asymptotic profile and the critical exponent for (7) with F = |u| p are the same of the corresponding heat equation with suitable initial data if = 0 [7][8][9]. A similar phenomenon appears when ∈ (0, 1∕2) but in this case two possible asymptotic profiles of anomalous diffusion appear; one profile is dominant if F = |u| p (see [10][11][12]) and the other profile is dominant if F = |u t | p (see [13]). On the other hand, the case ∈ (1∕2, 1] corresponds to a noneffective dissipation: the asymptotic profile of the solution to (7) contains oscillations analogous as the undamped wave equation (see [14][15][16][17][18][19]). However, the noneffective dissipation produces a better smoothing effect than the effective one. This is due to the fact that the noneffective dissipation produces a parabolic smoothing effect; for instance, the problem is well-posed in L p , for p ∈ [1, ∞] . On the other hand, the effective dissipation does not help to manage the regularity issues typical of the wave equation and other hyperbolic equations; for instance, the problem is well-posed in H s spaces, but not in L p spaces, when p ≠ 2.
The threshold case = 1∕2 inherits the benefits of the both effective and noneffective dissipations, and this allows to obtain the global (in time) existence of small data solutions in any space dimension n ⩾ 1 in the critical case (see [20]). The benefits of the both effective and noneffective dissipations are also gained in model (5), but with several important differences. The model in (1) describes the transition between the two regimens. For this reason, we expect that the critical exponents are the same as those in the effective case, namely p 0 and p 1 in (3) and (4), but the smoothing typical of the noneffective dissipation allows us to prove our result for any space dimension n ⩾ 1.
To prove the desired global (in time) existence results, we first derive long time decay estimates for the solution to the linear Cauchy problem with 0 ⩽ < 1∕2 < ⩽ 1 . We cannot directly follow the approach in [3] to derive the L 1 − L 1 estimate for (5) at high frequencies. In order to overcome this difficulty, we expand the kernel of solutions in a suitable way and we apply the Mikhlin-Hörmander multiplier theorem and Hardy-Littlewood theorem for the Riesz potential, to get some L r − L q estimates, with 1 < r ⩽ q < ∞.
This paper is organized as follows. In Sect. 2, we state our main results on global (in time) existence of small data solution and blow-up of solutions to (1). In Sect. 3, we prepare L r − L q low frequencies estimates and L m − L q high frequencies estimates for the linear Cauchy problem (8), where 1 ⩽ r ⩽ m ⩽ q ⩽ ∞ . Then, applying the derived estimates and Banach's fixed point theorem, we prove global (in time) existence results for (1) with |u| p or |u t | p , in Sect. 4. Eventually, in Sect. 5, the blow-up results in the subcritical case for (1) are derived.

Notation
We write f ≲ g when there exists a positive constant C such that f ⩽ Cg.

Main results
We first consider the case of power nonlinearity F = |u| p . where q = n∕(n − 2 ) , and If we replace the smallness of initial data in L 1 by the smallness of initial data in L , > 1 , the critical exponent becomes p 0 (n∕ , ) , as discussed in [13]. We demonstrate the global (in time) existence of energy solutions with small data in L , for supercritical and critical powers p ⩾ p 0 (n∕ , ). Theorem 2.2 Let n ⩾ 1 and 0 ⩽ < 1∕2 < ⩽ 1 . Let us fix ∈ (1, 2] , such that 2 < 1 if n = 1 , and assume where p 0 is defined in (3). Also we assume that p < n∕(n − 4 ) if n > 4 . Then, there exists a constant > 0 such that for any there is a uniquely determined energy solution to (1) with F = |u| p . Furthermore, the solution satisfies the following estimates: where q = n ∕(n − 2 ) , and Remark 2.1 Using Gagliardo-Nirenberg inequality, as a consequence of (12) and (13) in Theorem 2.1 or, respectively, (18) and (19) in Theorem 2.2, the solution to (1) with F = |u| p verifies the decay estimate for any q ∈ (q, ∞] if n < 4 , for any q ∈ (q, ∞) if n = 4 , and for any q ∈ (q, 2n∕(n − 4 )] if n > 4 (exception given for the case p = p 0 (n∕ , ) and = 1 ). On the other hand, interpolating (13) and (14) in Theorem 2.1 or, respectively, (19) and (20) in Theorem 2.2, the solution to (1) with F = |u| p verifies the estimate for any q ∈ [ ,q] . The latter is not a decay estimate. We mention that in the both cases, the estimate is the same of the solution to the linear problem (8), whose optimality is guaranteed by the diffusion phenomenon [10].

Remark 2.2
Actually, one may apply Theorem 2.2 to obtain the global (in time) existence result for (1) with F = |u| p and small initial data in L 1 ∩ L 2 for any p > p 0 (n, ) . Indeed, for a given p > p 0 (n, ) , it is sufficient to apply Theorem 2.2 with some ∈ (1, 2] such that p ⩾ p 0 (n∕ , ) . This strategy will be used to deal with (1) with F = |u t | p , to avoid the more challenging L 1 − L 1 estimates for u t . This difficulty is one peculiar difference with respect to the case ( , ) = (0, 1) studied in [3].
Thanks to Theorem 2.3, we immediately obtain the following. where p 1 is defined in (4). We also assume that p ⩽ n∕(n − 2 ) . Let us fix q ∈ (p, ∞) . Then, there exists > 0 such that for any and for any ∈ (1, q∕p] such that p ⩾ p 1 (n∕ , ) , there is a uniquely determined weak solution to (1) with F = |u t | p . Furthermore, the solution satisfies the estimates (24) and (25), for j = 0, 1.

Remark 2.3
If p ⩾ 2 , or if the initial datum is assumed to be small also in L 2 , it is not difficult to modify the proof of Theorem 2.3 or Corollary 2.1, and construct an energy solution in C [0, ∞), The optimality of the critical exponents p j (n∕ , ) in Theorems 2.1, 2.2, 2.3 and in Corollary 2.1 is guaranteed by the counterpart of nonexistence results in the subcritical and critical cases p ∈ (1, p j (n, )] , if = 1 , and in the subcritical case p ∈ (1, p j (n∕ , )) , if > 1 . Even if these results are easily derived following as in [13,21,22], using the test function method [23][24][25][26][27][28], we will sketch the outlines of the statement and its proof. Proposition 2.1 Let n ⩾ 1 and 0 ⩽ < 1∕2 < ⩽ 1 . Let us assume that u 1 ∈ L with ⩾ 1 verifies the following sign condition:

Estimates of solutions to the linear problem
Let us apply the partial Fourier transform with respect to x to (8), defining û(t, ) = F x→ (u(t, x)) . Then, we obtain the initial value problem The roots of the characteristic equation can be expressed by

Definition 3.1
We introduce three cut-off functions, namely int ( ), mid ( ) and for a sufficiently small > 0 and a sufficiently large N ≫ 1 , and such that In particular, we fix , N so that Ω , ⊂ Z mid (3 , N∕3).

Here and hereafter
The solution to (8) localized in Z mid ( , N) verifies the estimate for any 1 ⩽ r ⩽ q ⩽ ∞ , j + s ⩾ 0 , and t ⩾ 0 , for some C, c > 0 , independent of the datum. Indeed, mid is compactly supported in ℝ n ⧵ {0} and the real parts of the roots ± ( ) are negative in ℝ n ⧵ {0} . Thus, it will be sufficient to estimate the solution at low and high frequencies, that is, we will estimate int (D)u and ext (D)u. Let K 1 = K 1 (t, |x|) denote the fundamental solution to (8), that is,

Then, it holds
If we define then the roots and their difference can be expressed by To obtain the desired estimates at low frequencies, we rely on the following lemma.
where f and g satisfy the following conditions: , which denotes the integer part of (n + 3)∕2 . Then we have the following L r − L q estimates: in the following cases: Applying Lemma 3.1, the following L r − L q estimates follow.  (7) with F = 0 and = .

Proposition 3.1 Let u be the solution to
The optimality of these estimates is guaranteed by the diffusion phenomenon: the solution localized at low frequencies may be decomposed in two terms. Each term asymptotically behaves as the solution to an anomalous diffusion problem, namely where I 2 denotes the Riesz potential.
Proof We assume > 0 , since the case = 0 is easier; indeed, in this latter case, the term e + (| |)t produces an exponential decay. First let us consider j = 1 . In view of representation (31), we may apply Lemma 3.1, with g ± (| |) = − ± (| |) and with It is convenient to write By straightforward computation, the roots ± (| |) verify the estimates The above estimates are easily derived, due to the fact that neither ± nor f ± contain any oscillations, and that we are away from the zeroes of the root in (| |) , due to the fact that | | ⩽ . Namely, the estimates for the derivatives obey to the same rules of polynomials: each partial derivative with respect to produces a negative power of | | . This property is essential in the framework of the theory of L q bounded operators, see for instance, Mikhlin-Hörmander multiplier theorem. The desired estimate follows applying Lemma 3.1 with = 2s, 2s + 2 − 4 , exception given for the term related to f + (| |)e + (| |)t when s = 0 . In this latter case, we write and we separately apply Lemma 3.1 with = 0 and A = 1 , and with = 2 − 4 , to the two terms in the sum. Now let us turn to j = 0 . The proof follows by integration, if n(1∕r − 1∕q) < 2( − s) . Providing that n(1∕r − 1∕q) ⩾ 2( − s) , we set Due to the fact that the proof follows, applying Lemma 3.1 with = 2s − 2 , exception given for the case s = . In this latter case, we write and we separately apply Lemma 3.1 with = 0 and A = 1 , and with = 2 − 2 and = 2 − 4 , to the three terms in the sum. ◻

L m − L q estimates for high frequencies
At high frequencies, we write where To estimate the derivatives with respect to , we may proceed as we did at low frequencies in the proof of Proposition 3.1. Indeed, neither ± nor f ± contain any oscillations, and we are away from the zeroes of the root in (| |) , due to the fact that | | ⩾ N . For any ∈ ℕ n , we have the following estimates: Thanks to (36), we may easily derive L q − L q estimates, for q ∈ (1, ∞) , using Mikhlin-Hörmander multiplier theorem. Proof To prove our result, it is sufficient to show that for q ∈ (1, ∞) and for ⩾ 0 (we notice that the order of the Sobolev space in (39) is independent of j, if = 1 ). We define the multiplier So, the next representation holds: ). In particular, we notice that In an analogous way, we define and we may represent ). In this case, we may conclude We complete the proof of (38-39), applying Mikhlin-Hörmander multiplier theorem. ◻ Using Hardy-Littlewood-Sobolev theorem, we may extend Proposition 3.2 to L m − L q estimates with 1 < m < q < ∞. where I is the Riesz potential. Applying Theorem 3.2 with s + ∕2 in place of s, we can deriveâ The proof follows after noticing that, applying Hardy-Littlewood-Sobolev theorem, we may deduce Thus, the proof is complete. ◻ To derive L q − L q estimates for q = 1, ∞ , we cannot rely on Mikhlin-Hörmander multiplier theorem, so we prove that suitable multipliers associated with M ± , are bounded in L 1 showing that their inverse Fourier transforms belong to L 1 . Proof Using the decomposition (35), we would like to prove the following estimates: First of all, we fix j ⩾ k ⩾ 0 and s ⩾ 0 , and we consider the multiplier so that where a j,k,s = F −1 →x (â j,k,s (t, )) . To derive (45), we prove that We first consider |x| < 1∕N . As in [19], we use the identity to integrate by parts n − 1 times the function Since the boundary terms vanish since ext vanishes near {| | = N} , we obtain In particular, we can split each of the integrals of a j,k,s into two parts, which gives where Using (36) we find that for each exponent h > 0 , it holds which gives immediately for any k ⩽ j . Hence, we obtain Concerning I 2 , integrating by parts one more time, it yields Following a similar procedure as for (49), we can derive a j,k,s (t, ⋅) = 1 (2 ) n ∫ ℝ n e ix⋅â j,k,s (t, ) d .
We now consider the remaining case |x| ⩾ 1∕N . Using the identity (48) and integrating by parts n + 1 times, we may deduce The combination of (50) and (51) concludes the proof of (47), and then (45) follows.
In order to prove (46), we proceed as before, setting Then, we have Using (36) again we may observe that for each exponent h > 0 , it holds so that for any k ⩽ j . Repeating the procedure in the above, we arrive at This concludes the proof of our desired estimate (46). ◻

Summary of (L r ∩ L m ) − L q estimates
Thanks to Propositions 3.1-3.3 we may deduce the following estimates that will be used later to prove our global (in time) existence results.

Philosophy of our approach
Let us first introduce some notation for the proof of the global (in time) existence of small data solutions. Throughout this section, K 1 (t, x) denotes the fundamental solution to the linear Cauchy problem (29) with initial data u 0 = 0 and u 1 = 0 , where 0 is the Dirac distribution in x = 0 with respect to the spatial variables. As a consequence, we may represent the solution to the Cauchy problem (8)

in the form
We may introduce the operator where X(T) is an evolution space which will be defined in a suitable way in the proof of each theorem, and u non = u non (t, x) is an integral operator with the following representation According to the Duhamel's principle, we will prove the existence of the unique global (in time) solution to (1) as the fixed point of the operator N. Hence, in order to get the global (in time) existence and uniqueness of the solution in X(T), we need to prove the following two crucial estimates: with C > 0 , independent of T, where A denotes the space of the datum. As a consequence of Banach's fixed point theorem, the conditions (60) and (61) guarantee the existence of a uniquely determined solution u to (1) that is u solves the integral equation u = u lin + u non . We simultaneously gain a local and a global (in time) existence result.
Indeed, let R > 0 be such that CR p−1 < 1∕2 . Then N is a contraction on X R (T) = {u ∈ X(T) ∶ ‖u‖ X(T) ⩽ R} , thanks to (61). The solution to (1) is a fixed point for N, so if ‖u lin ‖ X(T) ⩽ R∕2 , then u ∈ X R (T) , thanks to (60). As a consequence, the uniqueness and existence of the solution in X R (T) follows by the Banach fixed point theorem on contractions. The condition ‖u lin ‖ X(T) ⩽ R∕2 is obtained taking initial data verifying ‖u 1 ‖ A ⩽ , with such that C ⩽ R∕2 . Since C, R and do not depend on T, the solution is global (in time).
In the proof of our global (in time) existence results, the following proposition will be useful, which use goes back to [29].

Proof of Theorem 2.1
Let us define the evolution space with its corresponding norm where we define We also recall that q = n∕(n − 2 ). Defining the data space A = L 1 ∩ L 2 , it follows immediately and so we conclude that u lin ∈ X(T) . Indeed, it is sufficient to apply (53) with q = 1 , (54) with q =q , r = 1 and m ∈ (1, min{q, 2}] , (56) with r = 1 , and (57) with r = 1.
In the remaining part of the proof we will estimate u non in the norm of X(T). To do this, let us introduce the following useful lemma.
Proof In the limit case q =q , the estimate (64) is an obvious consequence of ‖u‖ X(t) < ∞ . On the other hand, due to the restriction p ⩽ n∕(n − 4 ) if 4 < n , the Sobolev embedding H 2 ↪ L 2p holds, and (64) follows as a consequence of Gagliardo-Nirenberg inequality. ◻ Thanks to (53) with q = 1 , applying the Minkowski's integral inequality, we get Due to p > p 0 (n, ) = 1 + 2∕(n − 2 ) >q , we can apply (64) with q = p to get In particular, using again p > p 0 (n, ) , we obtain Therefore, applying Proposition 4.1 we conclude for t ⩽ T that In order to estimate u non (t, ⋅) in the L̄q space, we apply (54) with q =q , r = 1 and m ∈ (1, min{q, 2}] . Noticing that mp ⩽ 2p , we can apply (64) with q = p and with q = mp , to get where in the last inequality we used again (65) and Proposition 4.1.
In order to estimate ‖(−Δ) u non (t, ⋅)‖ L 2 we now apply (56) with r = 1 in [0, t/2] and with r = 2 in [t/2, t]. Using (64) with q = p and with q = 2p , noticing that we derive Here, we used (1 + t − s) ≈ (1 + t) for any s ∈ [0, t∕2] and (1 + s) ≈ (1 + t) for any s ∈ [t∕2, t] . Again, we plan to use (65). The first consequence is that On the other hand, due to the fact that ( − )∕(1 − ) ⩽ 1 , using (65) we get Summarizing the previous estimates, we proved the desired estimate To estimate the time derivative of the solution, we distinguish two cases. If 2 <q , that is, n < 4 and we may apply (57) with r = 1 to estimate using (64) with q = p and with q = 2p , (65) and Proposition 4.1. Now let 2 ⩾q , that is, n ⩾ 4 and We fix r * ∈ (1, 2) such that and we apply (57) with r = 1 in [0, t/2] and with r = r * in [t/2, t]. Then, we obtain where we applied (64) carrying q = r * p to obtain Again, we use (65) in two ways: to get (66), and to estimate so that we obtain the desired estimate Summarizing, we proved and this concludes the proof of (60).
In a similar way, we can prove (61). Indeed, (67) ‖u non ‖ X(T) ≲ ‖u‖ p X(T) , for j = 0, 1 and k = 0, 1 . We proceed as we did to prove (67), but we now use Hölder's inequality to estimate so that it is sufficient to replace ‖u‖

Proof of Theorem 2.2
We define the solution space For p > p 0 (n∕ , ) , we equip X(T) with the norm where we define Again, we recall that q = n ∕(n − 2 ). Let us take into consideration the case p = p 0 (n∕ , ) . In the case < 1 and 2 <q , we use the same norm as before. In the case < 1 and q ⩾ 2 , we modify the norm in the above, replacing (1 + t) ,2 ‖u t (t, ⋅)‖ L 2 with (1 + t) ,2 (log(2 + t)) −1 ‖u t (t, ⋅)‖ L 2 . Namely, ,q = min Considering p = p 0 (n∕ , ) and = 1 , we fix s ∈ [ , 1) such that so that H 2s ↪ L 2p and we modify the norm of X(T) in the following way: As in the proof of Theorem 2.1, defining the data space A = L ∩ L 2 , estimate (63) immediately follows, and so we conclude that u lin ∈ X(T) . Indeed, it is sufficient to apply (53) with q = , (54) with q =q , r = m = , (56) with r = , and (57) with r = .
It remains to estimate u non in the X(T) norm. Similarly to Lemma 4.1, the next lemma can be obtained. First, let p > p 0 (n∕ , ) . In this case, it is sufficient to follow the proof of Theorem 2.1, replacing the L 1 smallness of the initial datum by the L smallness.
For instance, thanks to (53) with q = , applying the Minkowski's integral inequality, one gets Since the fact that p > p 0 (n∕ , ) >q , we can apply (69) with q = p to get In particular, using again p > p 0 (n∕ , ) , we obtain (65) with n∕ in place of n, namely Hence, using Proposition 4.1 we conclude We omit the steps of the proof used to estimate u non (t, ⋅) in L q , and to estimate u non (t, ⋅) and u non t (t, ⋅) in L 2 , since they are analogous to those in the proof of Theorem 2.1. Now we consider the critical case p = p 0 (n∕ , ) in the following part. In this case, the strict inequality in (70) no longer holds, so we cannot follow the same proof of the supercritical case. For this reason, to deal with the critical case, we do not rely on estimate of |u(t, ⋅)| p in L , but in some L 1 spaces with 1 ∈ [1, ) . The same strategy is used in the context of semilinear damped waves in [30] and in the context of semilinear fractional diffusive equations in [31].
We fix 1 ∈ (1, ) , sufficiently close to to guarantee n(1∕ 1 − 1∕ ) < 2 and 1 p 0 (n∕ , ) ⩾q . As a consequence of Lemma 4.2, it yields where in the last equality we used p = p 0 (n∕ , ) . Using (54) with q = and r = m = 1 , we find where we used Proposition 4.1 with , < 1. Since 1 < , we find immediately Thus, using (54) with q =q and r = m = 1 , we obtain In order to estimate (−Δ) u non (t, ⋅) in the L 2 norm, we directly apply (56) with r = 1 in [0, t/2] and with r = 2 in [t/2, t]. Using (69) with q = 1 p and with q = 2p , noticing that we obtain the desired estimate In the case = 1 , we proceed in a similar way to estimate (−Δ) s u non (t, ⋅) in the L 2 norm, but we use (55) rather than (56), and the logarithmic term does not appear, due to the fact that s − < 1 − : We now distinguish two cases to estimate u non t . If 2 <q , that is, n < 4 ∕(2 − ) (this latter is true for any n ⩾ 1 if = 2 ), and we may apply (57) with r = 1 to estimate using (69) with q = 1 p and with q = 2p.
Now let 2 ⩾q , that is, n ⩾ 4 ∕(2 − ) and We fix r * ∈ ( , 2) such that and we apply (57) with r = 1 in [0, t/2] and with r = r * in [t/2, t]. Then, we obtain the desired estimate where we applied (69) with q = r * p and we used p = p 0 (n∕ , ) , to obtain Summarizing, we have proved (67), and this concludes the proof of (60). In a similar way, we prove (61) and we conclude the proof.

Proof of Theorem 2.3
First of all, we define the evolution space equipped with the corresponding norm Similarly to Lemmas 4.1 and 4.2, applying the interpolation of the norms of L and L p , we may easily prove the following lemma.

3 5 Proof of Proposition 2.1
We only sketch the proof of Proposition 2.1, which may be easily deduced using the test function introduced in [21,22] to modify Theorems 1, 2, 3, 4 in [13], so that they apply to fractional powers of the Laplace operator.
First, let j = 0 . Assume, by contradiction, that u is a global (in time) weak solution. Then it holds We fix = 2(1 − ) . Using Hölder's inequality and Young's inequality, taking K = 1 , we may estimate If > 1 , the sign assumption (28) implies that for R ≫ 1 , so that the contradiction follows from the inequality for large R and p < p 0 (n∕ , ) . If = 1 and p < p 0 (n, ) , then R n+2(1− )−2p � → 0 as R → ∞ and the inequality implies the contradiction, thanks to the sign assumption (27). If = 1 and p = p 0 (n, ) , we first deduce that I R is uniformly bounded, that is, u ∈ L p ([0, ∞) × ℝ n ) . As a consequence, and the contradiction follows for sufficiently large K ≫ 1 , thanks to (27), due to the fact that n − 2 p � = −2 < 0 for p = p 1 (n, ). This concludes the proof.

Concluding remarks
Initial data are assumed to be small in Sobolev spaces, since we take advantage of the diffusion phenomenon and of the smoothing effect to obtain the desired estimates for the solution.
These two crucial properties of equation (8) are related to the presence of the two dissipative terms. Assuming small, smooth initial data, possibly with compact support, could lead to gain more spatial regularity for the solution, at least for smooth nonlinearities, i.e., for large power nonlinearities. Still, we cannot expect a classical argument of well-posedness in C ∞ , due to the lack of finite speed of propagation, since the fractional Laplace operator is nonlocal. We did not investigate the possible gain of regularity of solution assuming more regular data and sufficiently large power nonlinearities, since in this paper we focused on some minimal data regularity which guarantees the existence of a global-in-time solution in the energy space, for any supercritical power.
The assumption u(0, x) = 0 in (1) may be easily replaced in Theorems 2.1, 2.2 and 2.3 by the smallness of the initial data u 0 (x) = u(0, x) in an appropriate space. For instance, we may supplement (11) in the statement of Theorem 2.1 with Here W 2,1 is the Sobolev space of functions in L 1 with their derivatives up to order 2. By the equivalence of the norm in Sobolev spaces and in Bessel potential spaces of positive even order, the smallness assumption in (78) implies, in particular, that This is sufficient to replace (63) in the proof of Theorem 2.1, by where now Similarly, in the statement of Theorem 2.2 and, respectively, Theorem 2.3 we may supplement (17) with and, respectively, On the other hand, the assumption u(0, x) = 0 plays a different role in the proof of Proposition 2.1. In particular, if j = 0 assuming that u(0, x) = u 0 (x) the identity (75) becomes (78) u 0 ∈ W 2,1 ∩ H 2 with ‖u 0 ‖ W 2,1 ∩H 2 ⩽ . u 0 , (−Δ) u 0 ∈ L 1 ∩ L 2 , ‖u 0 ‖ L 1 ∩L 2 + ‖(−Δ) u 0 ‖ L 1 ∩L 2 ⩽ .
If > 0 it is sufficient to assume u 0 ∈ L in order to estimate for both = and = , by using (74), and Hölder's inequality if > 1 . In this way, the presence of the first initial datum u 0 does not invalidate the argument in the proof of Proposition 2.1, in both the cases = 1 and > 1 . On the other hand, if = 0 then (27) and (28) must be, respectively, changed in the following assumptions: Under such assumptions one can prove in the case j = 0 the same results as in Proposition 2.1 with nonvanishing first initial datum. On the other hand, removing the initial datum assumption u 0 ≡ 0 in the case j = 1 raises more difficulties, whose investigation is beyond the scope of Proposition 2.1. (82)