Abstract
We consider the Cauchy problem for the nonlinear wave equation \(u_{tt} - \Delta _x u +q(t, x) u + u^3 = 0\) with smooth potential \(q(t, x) \ge 0\) having compact support with respect to x. The linear equation without the nonlinear term \(u^3\) and potential periodic in t may have solutions with exponentially increasing \(H^1(\mathbb {R}^{3}_{x})\) norm as \(t\rightarrow \infty \). In Petkov and Tzvetkov (IMRN, https://doi.org/10.1093/imrn/rnz014), it was established that by adding the nonlinear term \(u^3\), the \(H^1(\mathbb {R}^{3}_{x})\) norm of the solution is polynomially bounded for every choice of q. In this paper, we show that the \(H^k({{\mathbb {R}}}^3_x)\) norm of this global solution is also polynomially bounded. To prove this, we apply a different argument based on the analysis of a sequence \(\{Y_k(n\tau _k)\}_{n = 0}^{\infty }\) with suitably defined energy norm \(Y_k(t)\) and \(0< \tau _k <1\).
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1 Introduction
Consider for \(t \in {{\mathbb {R}}},\, x \in {{\mathbb {R}}}^3\), the Cauchy problem
where \(0 \le q(t, x) \in C^{\infty }({{\mathbb {R}}}_t \times {{\mathbb {R}}}^3),\,q(t, x) = 0\) for \(|x| \ge \rho > 0\) and
Set
For the Cauchy problem for the linear operator \(\partial _t^2u - \Delta _x u+ q(t, x) u \), there exist potentials \(q(t, x) \ge 0\) periodic in time with period \(T > 0\) such that for suitable initial data \(f = (f_1, f_2) \in {{\mathcal {H}}}({{\mathbb {R}}}^3) = H^1({{\mathbb {R}}}^3) \times L^2({{\mathbb {R}}}^3)\), we have
with \(C> 0, \, \alpha > 0\) (see [1, 2]). This phenomenon is related to the so called parametric resonance. On the other hand, adding a nonlinear term \(u^3\) for the Cauchy problem (1.1), there are no parametric resonances and for every potential q, the solution u(t, x) is defined globally for \(t \in {{\mathbb {R}}}\) and satisfies a polynomial bound
with constants \(B_0> 0, B_1 > 0\) depending on q and the initial data \(f \in {{\mathcal {H}}}\). This result has been obtained in [2, Theorem 2], and the proof was based on the inequality
where
In fact, the local Strichartz estimates and [2, Theorem 2] hold for every non-negative potential \(q(t, x) \in C^{\infty }({{\mathbb {R}}}_t \times {{\mathbb {R}}}^3)\) with compact support with respect to x satisfying the estimates (1.2) since in the proofs of these results the periodicity of q with respect to t is not used.
In this paper, we study the problem (1.1) with initial data \(f \in H^k({{\mathbb {R}}}^3) \times H^{k-1}({{\mathbb {R}}}^3),\, k \ge 2.\) Throughout the paper, we consider Cauchy problems with real-valued initial data f and real-valued solutions. First in Section 2, we establish a local result and we show the existence and uniqueness of the solution for \(t \in [s, s+ \tau _k]\) with initial data \(f \in H^k({{\mathbb {R}}}^3) \times H^{k-1}({{\mathbb {R}}}^3)\) on \(t = s\) and
where \(c_k\) depends on q and k (see Proposition 2.1). It is important to notice that \(\tau _k\) depends on the norm \(\Vert f\Vert _{{{\mathcal {H}}}}\) and since we have a global bound for the \({{\mathcal {H}}}\) norm of \((u, u_t)(t, x)\), the interval of local existence depends on the \({{\mathcal {H}}}\) norm of the initial data. We prove this result without using local Strichartz estimates. Next we show that the global solution in \({{\mathbb {R}}}\) is in \(H^k({{\mathbb {R}}}^3)\) for all \(t \in {{\mathbb {R}}}\) and the problem is to examine if the norm \(\Vert u(t, x)\Vert _{H^k({{\mathbb {R}}}^3)}, \,k \ge 2,\) is polynomially bounded. To do this, it is not possible to define a suitable energy \(Y_k(t) \ge 0\) involving
for which \(Y_k'(t) \le C_k Y_k^{\gamma _k}(t),\, 0<\gamma _k < 1.\) To overcome this difficulty, we follow another argument based on Lemma A.1 (see Appendix) which has an independent interest and apply local Strichartz estimates for the nonlinear equation. We first study the case \(k = 2\) in Section 4 and by induction, one covers the case \(k \ge 3\) in Section 5. Our principal result is the following
Theorem 1.1
For every potential q and every \(k \ge 2\), the problem (1.1) with initial data \(f \in H^k({{\mathbb {R}}}^3) \times H^{k-1}({{\mathbb {R}}}^3)\) has a global solution u(t, x) and there exist \(A_k > 0\) and \(m_k \ge 2\) depending on \(q, \,k\), and \(\Vert f\Vert _{{{\mathcal {H}}}}\) such that
We refer to [3] and the references therein for other results about polynomial bounds for the solutions of Hamiltonian partial differential equations. The method of the proof of Theorem 1.1 basically follows the approach in [3]. The main difficulty compared to [3] is that in our situation, we do not have a uniform bound on the \(H^1({{\mathbb {R}}}^3)\) norm and for that purpose, we need to apply the estimate of Lemma A.1 in the Appendix.
2 Existence and uniqueness of local solutions in \(H^k({{\mathbb {R}}}^3),\, k\ge 3\)
In this section, we study the existence and uniqueness of local solutions of the Cauchy problem
where \(f = (f_1, f_2) \in H^k({{\mathbb {R}}}^3) \times H^{k-1}({{\mathbb {R}}}^3),\, k\ge 1, 0< \tau < 1.\) We assume that \([s, s+ \tau ] \subset [0, a],\) where \(a > 1\) is fixed. The cases \(k = 1, 2\) have been investigated in [2, Section 3] by using the norms
For \(k=1\), the space \(S_0\) has been denoted as S. The number \(\tau \) is given by
with some positive constants \(c_1> 0, \gamma > 0\) depending on q. The case \(k \ge 3\) can be handled by a similar argument and we will show that with \(\tau =\tau _k\) defined by (2.2) with the constant \(c_1\) replaced by \(0 < c_k \le c_1\) depending on k and q, one has a local existence and uniqueness in the interval \([s, s+ \tau _k].\) Consider the linear problem
for \(t \in [s, s+ \tau _k]\) with \(u_0 = 0.\) For the solution of the above problem with right hand part \(- u_n^3\) and \(f = (f_1, f_2)\), we have a representation
Here \( U_0(t,s): {{\mathcal {H}}}\rightarrow {{\mathcal {H}}}\) is the propagator related to the free wave equation in \({{\mathbb {R}}}^3\) (see [2, Section 2]) and
To estimate \(\Vert u_{n+1}\Vert _{S_{k}},\) we apply the operator
Notice that this operator commutes with \(U_0(t- \tau )\) and \(\Vert U_0(t-s)\Vert _{{{\mathcal {H}}}\rightarrow {{\mathcal {H}}}} \le A\) for \(|t- s| \le 1\) with \(A > 0\) independent of k. Therefore
and
For \( A_k \tau \le 1/2 \) with \(A_k >0\), depending on k and q, the term involving \(Q(\tau )u_{n+1}\) in (2.4) can be absorbed by \(\Vert u_{n+1}\Vert _{S_k}\) and we deduce
Here and below the constants C depend on k and q and they may change from line to line but we will omit this in the notations. Next we define the norm
We will use the following product estimate
provided
For the proof of the classical estimate (2.5), we refer to [4]. We apply (2.5) with \(p = 2, q_1 = 3, q_2= 6, r_1 =6, r_2=3\) and get
For the term involving \(u_n^2\) in the above inequality, we apply the same estimate with \(p = 3, q_1= q_2= r_1= r_2 = 6\) and deduce
Consequently, by the Sobolev embedding theorem,
This implies
On the other hand, for the solution \(u_n\), we have the estimate
with some constant \(C_0 >0\) depending on q (see [2, Section 3]) and we deduce the bound
Thus choosing \(2CC_1\tau (2C_0)^2 \Vert (f_1, f_2)\Vert ^2_{{{\mathcal {H}}}} \le 1,\) we may prove by induction the estimate
Next, let \(w_n =u_{n+1} - u_n\) be a solution of the problem
By using the inequality
we can similarly show that
which implies the convergence of \((u_n)_{n \ge 0}\) with respect to the \(\Vert \cdot \Vert _{S_k}\) norm. Repeating the argument of [2, Section 3], we obtain local existence and uniqueness. Thus we get the following
Proposition 2.1
For every \(k \ge 1\), there exist \(C_k> 0,\,c_k > 0\), and \(\gamma > 0\) depending on q and k such that for every \((f_1,f_2) \in H^k({{\mathbb {R}}}^3) \times H^{k-1}({{\mathbb {R}}}^3)\), there is a unique solution \((u,u_t) \in C([s, s + \tau _k], H^k({{\mathbb {R}}}^3) \times H^{k-1}({{\mathbb {R}}}^3))\) of the problem (2.1) on \([s, s + \tau _k]\) with \(\tau _k = c_k( 1 + \Vert (f_1, f_2)\Vert _{{{\mathcal {H}}}})^{-\gamma }\). Moreover, the solution satisfies
It is important to note that for every k, \(\tau _k\) depends on the \({{\mathcal {H}}}\) norm of the initial data.
In [2], it was proved that one has a global solution \((u, u_t) \in C({{\mathbb {R}}}, {{\mathcal {H}}}({{\mathbb {R}}}^3))\) with initial data \((f_1, f_2) \in {{\mathcal {H}}}({{\mathbb {R}}}^3).\) It is natural to expect that for \((f_1, f_2) \in H^{k}({{\mathbb {R}}}^3) \times H^{k-1}({{\mathbb {R}}}^3)\), we have a global solution \((u, u_t) \in C({{\mathbb {R}}},H^k({{\mathbb {R}}}^3) \times H^{k-1}({{\mathbb {R}}}^3)).\)
Let \(a >1\) be fixed and let \(k \ge 1.\) We wish to prove that the global solution with initial data \(f \in H^{k+1}({{\mathbb {R}}}^3) \times H^{k}({{\mathbb {R}}}^3)\) is such that
According to [2, Theorem 2], for \(0 \le t \le a\), we have an estimate
where \(B_1 > 0\) and \(B_2 > 0\) depend only on \(\Vert f\Vert _{{{\mathcal {H}}}}.\) Consider
First for \(0\le t \le \tau _k(a)\), we apply Proposition 2.1. Next we apply Proposition 2.1 for the problem with initial data on \(t = \frac{2}{3}\tau _k(a)\) which is bounded by (2.7). Thus we obtain a solution in \([0, \frac{5}{3}\tau _k(a)]\) and we continue this procedure by step \(\frac{2}{3} \tau _k(a)\). On every step, the norm \(H^{k+1}({{\mathbb {R}}}^3) \times H^{k}({{\mathbb {R}}}^3)\) of \((u, u_t)\) will increase with a constant \(C_k\). Finally, if
we deduce
Hence, we established (2.8) and one has a bound of the \(H^{k+1}\times H^k\) norm. Since a is arbitrary, we obtain (2.8) for \(t \in {{\mathbb {R}}}\) and a global existence for \(t \in {{\mathbb {R}}}.\) In Section 5, we will improve (2.10) to polynomial bounds of the Sobolev norms.
3 Local Strichartz estimate for the nonlinear wave equation
Our purpose is to establish a local Strichartz estimate for the solution of the Cauchy problem
where \(f = (f_1, f_2) \in H^2({{\mathbb {R}}}^3) \times H^1({{\mathbb {R}}}^3),\, 0 < \tau \le 1.\) It is well known (see [2, Proposition 1]) that for the solution of the Cauchy problem
we have an estimate
where \(\frac{1}{p} + \frac{3}{r} = \frac{1}{2},\, 2 < p \le \infty .\) We will later choose \(r = \frac{4 + 2 \epsilon }{\epsilon }\) with \(0 < \epsilon \ll 1\) and this determines the choice of \(p > 2.\) For the solution of (3.1), we get
where we have used the estimate
Next, for the solution \(u(t, x) \in H^1({{\mathbb {R}}}^3)\) of (3.1) in \(]0, s+ \tau ]\) with initial data \(f = (u, u_t)(0, x) \in {{\mathcal {H}}}({{\mathbb {R}}}^3)\), we have a polynomial bound (see [2, Section 3])
where \(B_1> 0, B_2 >0\) depend only on \(\Vert f\Vert _{{{\mathcal {H}}}},\) and this implies
Now we will examine the continuous dependence on the initial data of the local solution to (2.1) given in Section 2. Let \(g_n = ((g_n)_1, (g_n)_2) \in H^{k+1}({{\mathbb {R}}}^3) \times H^k({{\mathbb {R}}}^3)\) be a sequence converging in \(H^k({{\mathbb {R}}}^3) \times H^{k-1}({{\mathbb {R}}}^3)\) to \(f = (f_1, f_2) \in H^k({{\mathbb {R}}}^3) \times H^{k-1}({{\mathbb {R}}}^3).\) Let
be the local solution of (3.1) with initial data \(g_n\). Setting \(v_n = w_n - u\), we obtain for \(v_n\) the equation
By the local Strichartz estimates for the linear equation with respect to \(v_n\), we get
This estimate for \(k = 1, 2\) has been proved in [2, Proposition 1]. The proof for \(k \ge 3\) follows the same argument. The constant \(C_k(a) > 0\) depends on k and on the interval [0, a], where \([s, s+ \tau ]\subset [0,a].\) In the notations below we will omit the dependence of the constants on k and a. Applying (2.5), we have
To handle \(P_n\), notice that the \(L^{\infty }([s, s+ \tau ], L^6({{\mathbb {R}}}^3))\) norms of u and \(w_n\), by local Strichartz estimates, can be estimated by \(\Vert f\Vert _{{{\mathcal {H}}}}\) and \(\Vert g_n\Vert _{{{\mathcal {H}}}}\). Therefore, for \(n \ge n_0\), we have
with a constant \(A_k\) depending on \(C_k(a)\) and \(\Vert f\Vert _{{{\mathcal {H}}}}\). Hence, we may absorb \(P_n\) by the left hand side of (3.5) choosing \(0 < \tau \le \frac{1}{2A_k}\) small. The analysis of \(Q_n\) is easy since we proved in [2, Subsection 3.2] that for all \(t \in [s, s+ \tau ]\), we have \(\Vert \nabla _x v_n(t, x)\Vert _{L^2({{\mathbb {R}}}^3)} \rightarrow 0\) as \(n \rightarrow \infty \) and the term in the braked \(\Bigl (...\Bigr )\) for \(t \in [0, a]\) is uniformly bounded with respect to n according to the analysis in Section 2 and estimate (2.10). Finally, we conclude that
4 Polynomial bound of the \(H^2({{\mathbb {R}}}^3)\) norm of the solution
Let \((u(t, x), u_t(t,x)) \in C([s, s + \tau ], H^2({{\mathbb {R}}}^3)) \times C([s, s+ \tau ], H^1({{\mathbb {R}}}^3)),\) where u(t, x) is the solution of the Cauchy problem (2.1) for \(t \in [s, s + \tau ]\). Taking the derivative \(\partial _{x_j} = \partial _j, \, j = 1, 2,3,\) and noting \(u_ j = \partial _j u,\, u_{j t} = \partial _{j}\partial _t u,\) one gets in the sense of distributions
It is easy to see that
In fact, our assumption implies that \(u(t, x) \in C([s, s + \tau ], L_x^{\infty }({{\mathbb {R}}}^3))\) and this yields \(u^2 u_j \in C([s, s + \tau ], L^2({{\mathbb {R}}}^3)).\) Therefore
Multiplying the equality (4.1) by \(u_{j t}\), we have
Assuming \((u(t, x), u_t(t, x)) \in C([s, s + \tau ], H^3({{\mathbb {R}}}^3) \times H^2({{\mathbb {R}}}^3))\), we can write
After an integration by parts in the integral \(\int \Delta _x (u_j) u_{j t} dx\) for solutions \((u(t, x), u_t(t, x)) \in C([s, s + \tau ], H^3({{\mathbb {R}}}^3) \times H^2({{\mathbb {R}}}^3))\), the equality (4.2) can be written as
where the derivative with respect to t of the left hand side is taken in the sense of distributions.
4.1 Justification of (4.3) for \((u(t, x), u_t(t, x)) \in C([s,s+ \tau ], H^2({{\mathbb {R}}}^3) \times H^1({{\mathbb {R}}}^3))\)
Introduce
Notice that the function X(t) is well defined. For the integral of \(u^2 u_j^2\), we have
Also a similar argument shows that the right hand side of (4.3) is well defined and it is a continuous function of t. For example,
This implies that the derivative with respect to t is taken in the classical sense. Now let \((g_n, h_n) \in H^3({{\mathbb {R}}}^3) \times H^2({{\mathbb {R}}}^3)\) converge to \((u(s, x), u_t(s, x))\) in \(H^2({{\mathbb {R}}}^3) \times H^1({{\mathbb {R}}}^3)\) as \(n \rightarrow \infty .\) Denote, as in Section 3, by \(w_n(t, x)\) the local solution of (3.1) with initial data \((g_n, h_n).\) Therefore, for \(t \in [s, s+ \tau ]\), we have
To justify these limits, we apply the estimates (4.4) and (4.5). For example,
and we use (3.6) for \(k= 2\). Passing to limit in the equality (4.3) for \(w_n\), we obtain it for u.
Consequently, after an integration with respect to t in (4.3), one deduces
4.2 Estimation of \(\int _{s}^{s +\tau }J_1(t)dt\)
Let \(0 < \epsilon \ll 1\) be a small number. First by the generalized Hölder inequality, one estimates
where
According to the estimate (2.7), for \(s \le t \le s + \tau ,\) by the local existence of a solution of (3.1) with initial data \( (u(s, x), u_t(s, x)) \in H^2({{\mathbb {R}}}^3) \times H^1({{\mathbb {R}}}^3)\) on \(t = s\), we obtain
with constant \(C_2 > 0\) depending on q. Next
Notice that we have a polynomial bound with respect to s for the norms \(\Vert u(s, x)\Vert _{H^1({{\mathbb {R}}}^3)}\) and \(\Vert u_t(s, x)\Vert _{L^2({{\mathbb {R}}}^3)}\) of the solution u(s, x) (see [2, Theorem 2]). Consequently, we obtain
where \(C_0> 0, C_1 > 0\) depend on \(\Vert (u,u_t)(0,x)\Vert _{H^1({{\mathbb {R}}}^3)}.\)
Now we pass to the estimate of \(\Vert u_t(t, x)\Vert _{L^{2+\epsilon }({{\mathbb {R}}}^3)}.\) By the Hölder inequality, we obtain
Hence, one deduces
Taking into account the above estimates for the integral with respect to t, one applies the Hölder inequality and for small \(\epsilon \), we have
where
To complete the analysis, we apply the Strichartz estimate (3.4) and deduce
Finally, for \(0 < \tau \le 1\) with \(y = 12\), we have
4.3 Estimation of \(\int _{s}^{s +\tau }I_1(t)dt\)
We apply a similar argument.
By the local existence result for \(t \in [s, s + \tau ]\), one has
and repeating the above argument, we deduce
4.4 Estimation of \(\int _{s}^{s +\tau }J_2(t)dt\)
It is easy to find a bound for this term since we have a polynomial estimate
and this yields
Combining (4.6), (4.7), (4.8), finally we get
4.5 Growth of \(H^2({{\mathbb {R}}}^3)\) norm
Let \(a > 1\) be a fixed number. According to [2] and Proposition 2.1, there exists a solution in \([s , s + \tau (a)] \subset [0, a]\) with initial data \(g \in H^2({{\mathbb {R}}}^3) \times H^1({{\mathbb {R}}}^3)\) on \(t = s\). Here
where \(c> 0, \gamma> 0, B_1> 0, B_2 > 0\) are independent of a and f. We choose \(N(a) \in {{\mathbb {N}}}\) so that \(a- \tau (a) < N(a)\tau (a) \le a.\) Setting \(X(n\tau (a)) = \alpha _n,\,n \le N(a),\) and exploiting (4.9), one deduces
We are in the position to apply Lemma A.1 in the Appendix and obtain
This estimate and the bound of the \(H^1({{\mathbb {R}}}^3)\) norm of the solution u(a, x) established in [2] imply a polynomial with respect to a bound of \(\Vert u(a, x)\Vert _{H^2({{\mathbb {R}}}^3)} + \Vert \partial _t u(a, x)\Vert _{H^1({{\mathbb {R}}}^3)}\). This implies the statement of Theorem 1.1 for \(k = 2.\)
5 Polynomial growth of the \(H^k({{\mathbb {R}}}^3)\) norm of the solution.
To examine the growth of the \(H^k({{\mathbb {R}}}^3)\) norm of the solution, we will proceed by induction. Assume that for \(1 \le s \le k-1, k \ge 3,\) we have polynomial bounds
for the global solution of the Cauchy problem of \(u_{tt} - \Delta _x u +qu+ u^3 = 0\) with initial data \((f_1, f_2) \in H^s({{\mathbb {R}}}^3) \times H^{s-1}({{\mathbb {R}}}^3).\) Consider the equality
with \(|\alpha | = k - 1.\) After an integration by parts which we can justify as in Section 4, we write
Clearly,
Applying (2.5) two times, one gets
and by the Sobolev theorem, \(\Vert u\Vert _{L^{\infty }({{\mathbb {R}}}^3)} \le C \Vert u\Vert _{H^2({{\mathbb {R}}}^3)}.\) Thus by our assumption,
Therefore, using the notation of Subsection 4.5 for \(n \tau (a) \le t \le (n+1) \tau (a),\) one deduces
On the other hand, applying (2.7), one obtains
The analysis of \(K_1(t)\) is easy and
Now define \(Y_k(t): = \Vert u(t, x)\Vert _{H^k({{\mathbb {R}}}^3)}^2 + \Vert \partial _t u(t, x)\Vert _{H^{k-1}({{\mathbb {R}}}^3)}^2\) and integrate the equality (5.1) from \(n\tau _k(a)\) to \((n+1)\tau _k(a)\) with respect to t, where \(0< \tau _k(a) < 1\) is defined by (2.9). Taking into account the above estimates, we have
Applying Lemma A.1 and repeating the argument of Subsection 4.5, we obtain a polynomial bound for \(Y_k(t)\) and this completes the proof of Theorem 1.1.
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Appendix
Appendix
The aim of this Appendix is to prove the following
Lemma A.1
Let \(\{\alpha _n\}_{n=0}^{\infty }\) be a sequence of nonnegative numbers such that with some constants \(0< \gamma <1\), \(C > 0\), and \(y \ge 0\), we have
Then there exists a constant \({\tilde{C}} > 0\) such that
Remark A.1
A similar estimate has been established in [3] for sequences \(\{\alpha _n\}\) satisfying the inequality
Proof
We can choose a large constant \(C_1 > 0\) such that
This implies with a new constant \(C_2 > 0\) the inequality
Setting \(\beta _n = \alpha _n + 1,\) we reduce the proof to a sequence \(\alpha _n\) satisfying the inequality
We will prove (A.1) by recurrence. Assume that (A.1) holds for \(n-1\). Therefore
To establish (A.1) for n, it is sufficient to show that for large \({\tilde{C}}\), one has
Setting \(C_2 {\tilde{C}}^{-\gamma } = \epsilon \), a simple calculus yields
Notice that since \(\frac{1}{2} \le 1- \frac{1}{n+ 1}\), we have
which implies
For small \(\epsilon > 0\), the right hand side of the above inequality is positive. Consequently, for the derivative, we have \(f'(n) > 0\) and one deduces
This completes the proof of (A.2). \(\square \)
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Petkov, V., Tzvetkov, N. Polynomial bounds on the Sobolev norms of the solutions of the nonlinear wave equation with time dependent potential. Arch. Math. 114, 71–84 (2020). https://doi.org/10.1007/s00013-019-01373-y
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DOI: https://doi.org/10.1007/s00013-019-01373-y