The generalized combined effect for one dimensional wave equations with semilinear terms including product type

We are interested in the so-called"combined effect"of two different kinds of nonlinear terms for semilinear wave equations in one space dimension. Recently, the first result with the same formulation as in the higher dimensional case has been obtained if and only if the total integral of the initial speed is zero, namely Huygens' principle holds. In this paper, we extend the nonlinear term to the general form including the product type. Such model equations are extremely meaningful only in one space dimension because the most cases in higher dimensions possess the global-in-time existence of a classical solution in the general theory for nonlinear wave equations. It is also remarkable that our results on the lifespan estimates are partially better than those of the general theory. This fact tells us that there is a possibility to improve the general theory which was expected complete more than 30 years ago.


Introduction
Let us consider the initial value problems; in R × (0, T ), u(x, 0) = εf (x), u t (x, 0) = εg(x), x ∈ R, ( where p, q, r > 1, A, B ≥ 0 and T > 0. We assume that f and g are given smooth functions of compact support and a parameter ε > 0 is "small enough".We are interested in the lifespan T (ε), the maximal existence time, of classical solutions of (1.1).Our results in this paper are the following estimates for A > 0 and B > 0; and Cε −(p+q)(r−1)/(r+1) for r + 1 2 ≤ p + q ≤ r, min{Cε −(p+q−1) , Cε −r(r−1)/(r+1) } otherwise if R g(x)dx = 0. ( Here we denote the fact that there are positive constants, C 1 and C 2 , independent of ε satisfying A(ε, C 1 ) ≤ T (ε) ≤ A(ε, C 2 ) by T (ε) ∼ A(ε, C).We note that (1.2) and (1.3) are already established in the special setting q = 0 by Morisawa, Sasaki and Takamura [17], but it is a non-trivial business to extend it to (1.1) due to the first term of product type for which different estimates from q = 0 are required in the proof.Also we note that the case of p = 1, or q = 1, is excluded because there is no hope to construct a classical solution due to lack of the differentiability.If we replace |u t | p |u| q with u t |u| q for p = 1 and q > 1, |u t | p u for p > 1 and q = 1, u t u for p = q = 1, then we may have the similar result at least of the existence part, but our method in this paper cannot be applicable directly for such terms.
This was verified by Zhou [23] for the upper bound with integer p, q satisfying p ≥ 1, q ≥ 0, p + q ≥ 2, and by Li,Yu and Zhou [12,13] for the lower bound with integer p, q satisfying p + q ≥ 2 including more general but smooth terms.Note that [23] is a preprint version of Zhou [22] in which only the case of q = 0 is considered.But it is easy to apply its argument to the case of q > 0 by making use of |u t | p |u| q = (p/(p + q)) p |u| (p+q)/p t p (1.4) as in [23].For the sake of completeness of this paper, we shall repeat its proof in Appendix below.
Therefore (1.2) and (1.3) are quite natural as taking the minimum of both results except for the first case in (1.3), in which, we have Cε −(p+q)(r−1)/(r+1) ≤ min{Cε −(p+q−1) , Cε −r(r−1)/(r+1) } for r + 1 2 ≤ p + q ≤ r. ( We shall call this special phenomenon by "generalized combined effect" of two nonlinearities.The original combined effect, which means the case of q = 0, was first observed by Han and Zhou [2] which targets to show the optimality of the result of Katayama [7] on the lower bound of the lifespan of classical solutions of nonlinear wave equations with a nonlinear term u 3 t + u 4 in two space dimensions including more general nonlinear terms.It is known that T (ε) ∼ exp (Cε −2 ) for the nonlinear term u 3 t and T (ε) = ∞ for the nonlinear term u 4 , but Katayama [7] obtained only a much worse estimate than their minimum as T (ε) ≥ cε −18 .Surprisingly, more than ten years later, Han and Zhou [2] showed that this result is optimal as T (ε) ≤ Cε −18 .They also considered (1.1) with q = 0 for all space dimensions n bigger than 1 and obtain the upper bound of the lifespan.Its counter part, the lower bound of the lifespan, was obtained by Hidano, Wang and Yokoyama [3] for n = 2, 3. See the introduction of [3] for the precise results and references.We note that the first case in (1.3) with q = 0 coincides with the lifespan estimate for the combined effect in [2,3] if one sets n = 1 formally.Indeed, [2] and [3] showed that T (ε) ∼ Cε −2p(r−1)/{2(r+1)−(n−1)p(r−1)} (1.6) holds for n = 2, 3 provided Later, Dai, Fang and Wang [1] improved the lower bound of lifespan for the critical case in [3].They also show that T (ε) < ∞ for all p, r > 1 in case of n = 1, i.e. (1.1) with q = 0.For the non-Euclidean setting of the results above, see Liu and Wang [14] for example, in which the application to semilinear damped wave equations is included.
Finally we strongly remark that our estimates in (1.2) and (1.3) are better than those of the general theory by Li, Yu and Zhou [12,13] in case of r + 1 2 < p + q < r and R g(x)dx = 0 with integer p, q, r ≥ 2. Because our result on the lower bound of the lifespan can be established also for the smooth terms as u tt − u xx = u p t u q + u r .The typical example is (p, q, r) = (2,2,6).This fact shows a possibility to improve the general theory.For details, see the last half of the next section.We note that this kind of observations in Morisawa, Sasaki and Takamura [17] has an error by wrong citation in the third case in (2.24) in [17].This paper corrects it.We also note that, even for the original combined effect of q = 0, the integer points satisfying (1.7) are (p, r) = (2, 3), (3,3), (3,4) for n = 2 and (p, r) = (2, 2) for n = 3, but (1.6) with p = r agrees with the case of A = 0 and B > 0. See Introduction of Imai, Kato, Takamura and Wakasa [4] for references on the case of A = 0 and B > 0. Hence one can say that only the lifespan estimates with (p, r) = (2, 3), (3,4) for n = 2 are essentially in the combined effect case.If q = 0, p is replaced with p + q in the results above.
Therefore it has less meaningful to consider (1.1) in higher space dimensions, n ≥ 2, if we discuss the optimality of the general theory.
Of course, some special structure of the nonlinear terms such as "null condition" guarantees the global-in-time existence.See Nakamura [16], Luli, Yang and Yu [15], Zha [19,20] for examples in this direction.But we are interested in the optimality of the general theory.The details are discussed at the end of Section 2 below.This work is initiated by series of papers, Kitamura [8], Kitamura, Morisawa and Takamura [9,10], Kitamura, Takamura and Wakasa [11], in which the weighted nonlinear terms are considered for the purpose to be a trigger to extend the general theory to the one for non-autonomous equations.
This paper is organized as follows.In the next section, the preliminaries are introduced.Moreover, (1.2) and (1.3) are divided into four theorems, and we compare our results with those of the general theory.Sections 3 is devoted to the proof of the existence part of (1.2).Sections 4 and 5 are devoted to the proof of the existence part of (1.3).Their main strategy is the iteration method in the weighted L ∞ space due to Morisawa, Sasaki and Takamura [17] which is originally introduced by John [5].Finally, we prove the blow-up part of (1.2) and (1.3) by following essentially, Han and Zhou [2] for the generalized combined effect, and the iteration argument in [17] for other cases.

Preliminaries and main results
Throughout this paper, we assume that the initial data (f, g) Let u be a classical solution of (1.1) in the time interval [0, T ].Then the support condition of the initial data, (2.1), implies that For example, see Appendix of John [6] for this fact.
It is well-known that u satisfies the following integral equation.
where u 0 is a solution of the free wave equation with the same initial data, and a linear integral operator L for a function v = v(x, t) in Duhamel's term is defined by Then, one can apply the time-derivative to (2.3) to obtain where L ′ for a function v = v(x, t) is defined by On the other hand, applying the space-derivative to (2.3), we have and where L ′ for a function v = v(x, t) is defined by Therefore, u x is expressed by u and u t .Moreover, one more space-derivative to (2.6) yields that Similarly, we have that Therefore, u tt is expressed by u, u t , u x , u tx and so is u xx , because of . First, we note the following fact.
with some T > 0.Then, there exists a positive constant 1) exists as far as T satisfies where 0 < ε ≤ ε 1 , and c is a positive constant independent of ε.
Theorem 2.2 Let A > 0 and B > 0. Assume (2.1) and R g(x)dx = 0. (2.17) Then, there exists a positive constant 1) exists as far as T satisfies where 0 < ε ≤ ε 2 , and c is a positive constant independent of ε.Then, there exists a positive constant ε 3 = ε 3 (f, g, p, q, A, B, R) > 0 such that any classical solution of (1.1) in the time interval [0, T ] cannot exist as far as T satisfies where 0 < ε ≤ ε 3 , and C is a positive constant independent of ε.
On the other hand, we have that Moreover, we see that Therefore Theorem 2.2 and Theorem 2.4 imply (1.3).
The proofs of four theorems above appear in the following sections.Form now on, we shall compare our results with those of the general theory by Li, Yu and Zhou [12,13], in which the following problem of general form is considered: where we denote D := (∂ t , ∂ x ) and F ∈ C ∞ (R 5 ) satisfies Then, the lifespan of the classical solution of (2.23) defined by T (ε) has estimates from below as in general, This is the result of the general theory.If one applies it to our problem (1.1) with F (u, Du, ∂ x Du) = u p t u q + u r with p, q, r ∈ N, (2.25) one has the following estimates in each cases.
• When p + q ≥ r, then, similarly to the case above, we have to set α = r − 1, which yields that We note that the third case does not hold for (2.25) by ∂ r u F (0) = 0.In conclusion, for the special nonlinear term in (2.25), the result of the general theory is Therefore a part of our results, is better than the lower bound of T (ε) because of If one follows the proof in the following sections, one can find that it is easy to see that our results on the lower bounds also hold for a special term (2.25) by estimating the difference of nonlinear terms from above after employing the mean value theorem.We note that we have infinitely many examples of (p, q, r) = (m, m, 2m + 1) as the inequality holds for m = 2, 3, 4, . ... This fact indicates that we still have a possibility to improve the general theory in the sense that the optimal results in (2.26) should be included at least.

Proof of Theorem 2.1
We basically employ the argument in Morisawa, Sasaki and Takamura [17] here.According to Proposition 2.1, we shall construct a C 1 solution of (2.14).
Let {(u j , w j )} j∈N be a sequence of Then, in view of (2.9) and (2.12), ((u j ) x , (w j ) x ) has to satisfy so that the function space in which {(u j , w j )} converges is where First we note that supp (u j , w It is easy to check this fact by assumption on the initial data (2.1) and the definitions of L, L, L ′ , L ′ in the previous section.
The following lemma contains some useful a priori estimates.
Then there exists a positive constant C independent of T and ε such that Proof.The proof of Proposition 3.1 is completely same as the one of Proposition 3.1 in Morisawa, Sasaki and Takamura [17] because u 1 has no weight, so that w p 2 in [17] is simply replaced with w p 2 u q 1 .✷ Let us continue to prove Theorem 2.1.Set The convergence of the sequence {(u j , w j )}.
Making use of and together with Proposition 3.1, we have that and similarly Here we employ Hölder's inequality to obtain and so on.Therefore the convergence of {u j } follows from are fulfilled.
The convergence of the sequence {((u j ) x , (w j ) x )}.
On the other hand, we have that The first term on the right hand side of this inequality is divided into three pieces according to Since one can employ the estimate and the same one in which w is replaced with u, we obtain that Hence it follows from (3.6) and (3.10) that as j → ∞.Therefore we obtain the convergence of {((u j ) x , (w j ) x )} provided pAC(3Mε Continuation of the proof. The convergence of the sequence {(u j , w j )} to (u, w) in the closed subspace of X satisfying is established by (3.5), (3.7), (3.9), and (3.11), which follow from where Therefore the statement of Theorem 2.1 is established with In this section also, we basically employ the argument in Morisawa, Sasaki and Takamura [17].But the different estimates for the product term are required here.First we note that the strong Huygens' principle holds in this case of (2.17), where This is almost trivial if one takes a look on the representation of u 0 in (2.4) and the support condition on the data in (2.1).But one can see it also by Proposition 2.2 in Kitamura, Morisawa and Takamura [9] for the details.So, our unknown functions are U := u − εu 0 and ) so that the function space in which {(U j , W j )} converges is where and χ D is a characteristic function of D. Similarly to the proof of Theorem 2.1, we note that supp (U j , W The following lemmas are a priori estimates in this case.
Then there exists a positive constant E independent of T such that where p − m, q − m > 0 (m = 0, 1, 2) and the norm • ∞ is defined by Proof.This lemma is exactly same as Proposition 5.1 in Morisawa, Sasaki and Takamura [17].✷ We note that U 0 in the theorem above will be replaced with u 0 or u 0 t and their spatial derivatives later due to (4.1).
Then there exists a positive constant C independent of T such that The proof of Proposition 4.2 is established in the next section.
Let us proceed the proof of Theorem 2.2.Set where E is the one in (4.4).We note that The four quantities, ε 2i (i = 1, 2, 3, 4), are defined in the following; and where and Moreover, we set where ε 21 is the one in (4.6) and ε 23i (i = 1, 2, 3, 4) are defined by . Finally, we also set (4.9) The convergence of the sequence {(U j , W j )}.It follows from (4.2), Proposition 4.1 and 4.2 that and similarly Hence the boundedness of {(U j , W j )}, i.e.
Continuation of the proof.
The convergence of the sequence {(U j , W j )} to (U, W ) in the closed subspace of Y satisfying U 3 , (U x ) 3 , W 4 , (W ) x 4 ≤ 5Nε min{p+q,r} is established by (4.6), (4.7), (4.8), (4.9), (4.13), (4.16), (4.22) and (4.26).Therefore the statement of Theorem 2.2 is established with In this section, we prove a priori estimate (4.5).Note that three estimates with |U| r are already obtained by Proposition 5.2 in Morisawa, Sasaki and Takamura [17], so that we shall prove other three estimates with |W | p |U| q .Here a positive constant C independent of T and ε may change from line to line.
It follows from the assumption on the supports and the definition of where the integrals J + and J − are defined by respectively.First we note that it is sufficient to estimate J ± for x ≥ 0 due to its symmetry, J + (−x, t) = J − (x, t).
For (x, t) ∈ D ∩ {x ≥ 0}, we have It is trivial that J ± (x, t) ≤ C for t + x ≤ R. Therefore we obtain the third inequality in (4.5).The fifth inequality in (4.5) readily follows from the computations above.The proof of Proposition 4.2 is now completed.✷ 6 Proofs of Theorem 2.3 and Theorem 2.4 The essential argument to obtain the upper bound of the lifespan is the following.Let u be a classical solution of (1.1) in a time interval [0, T ].If T is bigger than some quantity depending on ε, we will meet a contradiction to the fact that u is a classical solution.This situation gives us that the lifespan should be less than the quantity due to its definition.
Proof of Theorem 2.3.
Neglecting the second term of our equation as , we have that there exists a constant ε 31 = ε 31 (f, g, p, q, A, R) such that the contradiction appears provided holds for 0 < ε ≤ ε 31 and some positive constant C 31 independent of ε.
Because it is already obtained by Zhou [23] for the equation in which it is trivial that "=" can be replaced with "≥ A×".As stated in Introduction, we shall repeat its proof in Appendix below.By virtue of the same reason and Zhou [21], making use of we have that there exists a constant ε 32 = ε 32 (f, g, r, B, R) such that the contradiction appears provided holds for 0 < ε ≤ ε 32 and some positive constant C 32 independent of ε.Therefore, taking ε 3 = min{ε 31 , ε 32 , 1} and C = min{C 31 , C 32 }, we have the desired lifespan estimate by (6.1) and (6.2).
Set F (t) := On the other hand, neglecting the second term of the integrand, we have that |u t (x, t)| p |u(x, t)| q dx for t ≥ R/2.Then it follows from (6.5) that Integrating this inequality and employing the fact that F ′′ (t) ≥ 0 for t ≥ 0 and for 1 < r < 3 in this case, (r + 1)/2 ≤ p + q ≤ r.Therefore it is possible to take T 2 = T 0 in Lemma 2.2 in [18], so that there exists a constant ε 43 = ε 42 (f, p, q, r, A, B, R) such that the contradiction appears provided The proof of Theorem 2.4 is completed now.✷
Then, w ≡ u t in R × [0, T ] holds and u is a classical solution of (1.1) in R × [0, T ].