Existence of weak solutions to the Cauchy problem of a semilinear wave equation with supercritical interior source and damping

In this paper we show existence of finite energy solutions for the Cauchy problem associated with a semilinear wave equation with interior damping and supercritical source terms. The main contribution consists in dealing with super-supercritical source terms (terms of the order of $|u|^p$ with $p\geq 5$ in $n=3$ dimensions), an open and highly recognized problem in the literature on nonlinear wave equations.

We are interested in the existence of weak solutions to (SW) on the finite energy space H 1 (R n ) × L 2 (R n ). We will work with the following notation : | · | p,Ω denotes the L p (Ω) norm, while for the L 2 norm we simply use | · | Ω ; when there is no danger of confusion we simplify the notation | · | p,Ω to | · | p .
For the sake of exposition, we will focus on the most relevant case of dimension n = 3, but the analysis can be adapted to any other value of n. In this case, we classify the interior source f based on the criticality of the Sobolev's embedding H 1 (R 3 ) → L 6 (R 3 ) as follows: (i) subcritical: 1 ≤ p < 3 and critical: p = 3. In these cases, f is locally Lipschitz from H 1 (R 3 ) into L 2 (R 3 ); (ii) supercritical: 3 < p < 5. For this exponent f is no longer locally Lipschitz, but the potential well energy associated with f is still well defined on the finite energy space; (iii) super-supercritical: 5 ≤ p < 6. The source is no longer within the framework of potential well theory, due to the fact that the potential energy may not be defined on the finite energy space.
1.1. Assumptions. Throughout the paper we will impose the following conditions on the source and damping terms: (A g ) g is increasing and continuous with g(0) = 0. In addition, the following growth condition at infinity holds: there exist positive constants l m , L m such that for |s| > 1 we have l m |s| m+1 ≤ g(s)s ≤ L m |s| m+1 with m ≥ 0.
Remark 1. Note that the Assumption (A f ) allows for both types of supercriticality.
Also, (A f ) guarantees that f is locally Lipschitz from is an open connected set with smooth boundary ∂Ω. Let f and g be two real valued functions f and g which satisfy (A f ) and (A g ), and further suppose that u 0 ∈ H 1 0 (Ω) ∩ L p+1 (Ω) and A weak solution on Ω T of the boundary value problem is any function u satisfying ). Remark 2. A weak solution for the Cauchy problem (SW) is defined by taking in the above definition Ω = R n with no boundary conditions. 1.2. Relationship to previous literature. Significance of results. Semilinear wave equations with interior damping-source interaction have attracted a lot of attention in recent years. In the case of subcritical source f , local existence and uniqueness of solutions are standard and they follow from monotone operator theory [1]. In [8], the authors considered the case of polynomial damping and source, i.e. g(u t ) = |u t | m−1 u t and f (u) = |u| p−1 u and showed that if the damping is strong enough (m ≥ p), the solutions live forever, while in the complementary region m < p, the solutions blow-up in finite time. For supercritical interior sources, [2], [7] and [15] exhibited existence of weak solutions for a bounded domain Ω, under the restriction p < 6m/(m + 1), while [17] obtained the same results for Ω = R 3 , and compactly supported initial data, with p < 6m/(m + 1). In this case, it was shown additionally by [14] that if the interior damping is absent or linear, the exponent p may be supercritical, i.e. p < 5; also, in [14] the initial data may not be compactly supported. The case of super-supercritical sources on a bounded domain was analyzed and resolved recently in [3], [4], [5]. The authors considered the wave equation with interior and boundary damping and source interactions, and proved existence and uniqueness of weak solutions. Moreover, they provided complete description of parameters corresponding to global existence and blow-up in finite time. We will provide more details on these results in the next section.
Our paper provides existence of solutions to wave equations on R 3 for the case of super-supercritical sources. The method used will also provide an alternative proof in the case of supercritical (and below) interior sources. Thus our paper extends the known existence results to the super-supercritical case (we include the dark shaded regions 5 ≤ p < 6). We summarize our results and improvements over previous literature with the following illustration: Note that for the range of exponents m ≥ p (region I above) one expects global existence of solutions, while for m < p the solutions may blow up in finite time (according to the preliminary results of [8,2,3,5] obtained on bounded domains).

Preliminaries
We include in this section the following theorems which were proved in [14] and which will be used in the proof of our main result.
admits a unique solution u on the time interval [0, T ] in the sense of the Definition 1.1, i.e., u ∈ C(0, T ; H 1 0 (Ω)) ∩ L p+1 (Ω T ), u t ∈ L 2 (Ω T ) ∩ L m+1 (Ω T ). The finite speed of propagation property is known to hold for wave equations with nonlinear damping and/or with source terms of good sign, i.e. their contribution to the energy of the system is decretive. The following theorem states that the property remains true for source terms of arbitrary sign, as long as they are Lipschitz (for a proof see [14]). (1) if the initial data u 0 , u 1 is compactly supported inside the ball B(x 0 , R) ⊂ Ω, then u(x, t) = 0 for all points x ∈ Ω outside B(x 0 , R + t); (2) if (u 0 , u 1 ), (v 0 , v 1 ) are two pairs of initial data with compact support, with the corresponding solutions u(x, t), respectively v(x, t), and u 0 ( We conclude this section by stating the following result which appears in [3,4] and whose analog on R 3 we will prove in the next section. (1) , and l m given by (A g ).
Remark 3. The condition |f (s)| ≤ C|s| p−2 is needed for the uniqueness, but not for the existence of solutions.

Local in time existence of solutions to the Cauchy problem
Our main result states: and consider the Cauchy problem Proof. We identify the following steps: 3.1. Local existence on bounded domains. Consider for now the problem (SWB) where Ω is an open, bounded domain with smooth boundary. First we will solve the existence problem on such a domain; in the second step we will cut the initial data in pieces defined on small domains; finally, we will show how to piece together the solutions defined on these small domains to obtain existence of solutions on the entire space R 3 .
Approximation of f : We consider the following approximation of equation (SWB), with n → ∞ as the parameter of approximation: . We construct the approximating functions f n as follows: let η be a cutoff smooth function such that: . This means that , |u| ≤ n f (u)η(u) , n < |u| < 2n. 0 , otherwise.
In the sequel we will use the notationm = m+1 m . In order to prove the claim, we consider the following three cases: Case 2: n ≤ |u| , |v| ≤ 2n. Then we have the following computations: Now using the definition of the cutoff function η and the fact that |v| ≤ 2n, we can see that |v| max ξ |η (ξ) ≤ C and thus (3) becomes For the second term on the right side of (4), we use Hölder's Inequality with p and p/(p − 1), the fact that p(m + 1)/m ≤ 6/(1 + 2ε), and Sobolev's Imbedding which proves that f n is locally Lipschitz H 1−ε (Ω) → L m+1 m (Ω).
Case 3: If |u| ≤ n and n < v ≤ 2n, then we have In (6), we can replace 1 = η(u), since |u| ≤ n and then the calculations follow exactly as in case 2.
By using the regularity properties of the solutions u n , we apply the energy identity to the "n"-problem and obtain that for each 0 < T < T max , we have |u n t (0)| 2 Ω + |∇u n (0)| 2 Ω . A-priori bounds: Remember the assumptions on g and f : • g(s)s ≥ l m |s| m+1 for |s| ≥ 1 • f n is locally Lipschitz: Going back to (12), we estimate the terms involving the source f n by using Hölder's Inequality withm = m+1 m and m + 1, followed by Young's Inequality with the corresponding components. For simplicity, in the following computations we use u(t) instead of u n (t) .
Combining (12) with (13) and using the growth conditions imposed on g, we obtain: Choosing ε 1 < lm 2 and since m > 1, we obtain that for all T < T max , we have (14) |u n t (T )| 2 Ω + |∇u n (T )| 2 Ω ≤ [|u n t (0)| 2 Ω + |∇u n (0)| 2 Ω + CT ] · e C lm,K T , where C = C(g, f, ε 1 , m) and C lm = C ε1 Lm f (K). Also, we have From (15), combined with the growth assumptions imposed on the damping g, we obtain that Therefore, on a subsequence we have (u n , u n t ) → (u n , u n t ) weakly in H 1 (Ω) × L 2 (Ω) u n t → u t weakly in L m+1 (0, T ; Ω) g(u n t ) → g * weakly in L m+1 m (0, T ; Ω) , for some g * ∈ L m+1 m (0, T ; Ω). We want to show that g * = g(u t ). In order to do that, consider u m and u n be the solutions to the approximated problem corresponding to the parameters m and n. For sake of notation, letũ(t) = u n (t) − u m (t) andũ t (t) = u n t (t) − u m t (t). Then from the energy identity we obtain that for any T < T max we have First we will show that Ω T (f n (u n (t))−f m (u m (t)))(ũ t (t)) dxdt → 0 as m, n → ∞.
Recall thatm = m+1 m and | · | s = | · | L s (Ω) . Using Hölder's Inequality withm and m + 1, we obtain: Now we use the fact that f is locally Lipschitz H 1−ε (Ω) → Lm(Ω) and obtain: We know that u n (t) → u(t) weakly in H 1 (Ω) and since the embedding H 1−ε (Ω) ⊂ H 1 (Ω) is compact, we get that u n (t) → u(t) strongly in H 1−ε (Ω). We also know that |f n (u)−f (u)|m → 0 as n → ∞ (and same for m) and that |u n t (t)−u m t (t)| m+1,Ω ≤ C for t < T max . Thus from (18) we obtain the desired result Ω T [f n (u n (t)) − f m (u m (t))]ũ t (t) dxdt → 0 as m, n → ∞. Now we let m, n → ∞ in (17) and remembering that g is monotone, we obtain: Cutting the initial data. Consider now a pair of initial data (u 0 , u 1 ) ∈ H 1 (R 3 ) × L 2 (R 3 ) and let K be an upper bound on the energy norm of the initial data, more precisely take K such that We find r such that where ω 3 is the volume of the unit ball in R 3 and C * is the constant from the Sobolev inequality (which does not depend on x 0 nor r). It can be easily shown that the above inequalities are satisfied by r chosen such that The fact that r can be chosen independently of x 0 is motivated by the equiintegrability of the functions u 0 , ∇u 0 , u 1 . For each of the functions u 0 , ∇u 0 , u 1 we apply the following result of classical analysis: If f ∈ L 1 (A), with A a measurable set, then for every given ε > 0, there exists a number δ > 0 such that E |f (x)|dx < ε, for every measurable set E ⊂ A of measure less than δ (see [6]).

3.3.
Patching the small solutions.
The key argument that we use in order to construct the solution to the Cauchy problem from the "partial" solutions to the boundary value problems set on the balls B(x 0 , r) constructed in section 3.2 uses an idea due to Crandall and Tartar. They first used this type of argument to obtain global existence of solutions for a Broadwell model with arbitrarily large initial data starting from solutions with small data (see [18]). Subsequently, the second author has recast it in the framework of semilinear wave equations and showed local existence of solutions for (SW) on the entire space R 3 (see [14]); the argument may also be employed on bounded domains as it was done in [15].
Step 1. Construction of partial solutions. Consider a lattice of points in R 3 denoted by x j situated at distance d > 0 from each other, such that in every ball of radius d we find at least one x j . Next construct the balls B j := B(x j , r/2), where r is given by (22) and inside each B j take a snapshot of the initial data. More precisely, construct (u xj 0 , u xj 1 ) by the procedure used in subsection 3.2. On each of the balls B(x j , r) we use theorem 2.1 for the approximated problem given by the system (2) to obtain existence of solutions u xj ,n up to a time T (K) independent of x j and of n. These solutions will satisfy the estimate (12) on B(x j , r + T (K)). Following the arguments from section 3.1 we pass to the limit in the sequence of approximations u xj ,n on each of the balls B j and obtain a solution u xj .
Step 2. Patching the small solutions. For j ∈ N let For every set of intersection I j,l := C j ∩ C l the maximum value for time contained in it is equal to (r − d)/2 (see figure above). For t < r/2 we define the piecewise function: This solution is defined only up to time (r − d)/2, since the cones do not cover the entire strip R 3 × (0, r/2). By letting d → 0 we can obtain a solution well defined up to time r/2. Thus, we have u defined up to time r/2, which is the height of all cones C j . Every pair (x, t) ∈ R 3 × (0, r/2) belongs to at least one C j , so in order to show that this function from (26) is well defined, we need to check that it is single-valued on the intersection of two cones. Also, we need to show that the above function is the solution generated by the pair of initial data (u 0 , u 1 ). Both proofs will be done in the next step.
Step 3. The solution given by (26) is valid. To show that u defined by (26) is a proper function, we use the same result of uniqueness given by the finite speed of propagation. First note that for n ≥ 3 the intersection I j,l is not a cone, but it is contained by the cone C j,l with the vertex at ((x j + x l )/2, (r − d)/2) of height (r − d)/2. In this cone we use the uniqueness asserted by the finite speed of propagation as follows. Recall that the approximations f n are Lipschitz inside the balls {B(x j , r)} j∈R 3 , hence the finite speed property holds for the solutions u j,n . First note that the cones C j,l contain the sets I j,l , but C j,l ⊂ C j ∪ C l . In C j and C l we have the two solutions u j,n and u l,n hence, in C j,l we now have defined two functions which can pose as solutions. Since u j,n and u l,n start with the same initial data ((u j 0 , u j 1 ) = (u 0 , u 1 ) = (u l 0 , u l 1 ) on B j ∩ B l ), hence they are equal in C j,l ; since I j,l ⊂ C j,l we proved u j,n = u l,n on I j,l . By letting n → ∞ we get u j = u l on I j,l . Therefore, u is a single-valued (proper) function.
Finally, the fact that this constructed function u is a solution to the Cauchy problem (SW) is immediate since it satisfies both, the wave equation and the initial conditions. Remark 4. (on global existence) The above method of using cutoff functions and "patching" solutions based on the finite speed of propagation property will work the same way in the case when we have global existence on bounded domains. Since we can choose the height of the the cones as large as we wish the solutions exist globally in time.

Remark 5. (on uniqueness)
In [4] the authors showed under the same assumptions (A f ), (A g ) that the boundary value problem (SWB) admits a unique solution (in fact, the result was shown in the presence of damping and source terms in the interior and on the boundary. The methods employed here seem to preclude us from obtaining a corresponding result for the Cauchy problem (SW) since they are obtained by passing to the limit in sequences of approximations.