Asymptotic profile of solutions to the semilinear beam equation

In this paper, we investigate the initial value problem for the semilinear beam equation. Under a small condition on the initial value, we prove the global existence and optimal decay estimate of solutions. Moreover, we show that as time tends to infinity, the solution is asymptotic to a diffusion wave, which is given explicitly in terms of the solution of parabolic equation.

MSC:35L30, 35L75.

We investigate the initial value problem for the following semilinear beam equation:

with the initial value

Here u=u(x,t) is the unknown function of x\in \mathbb{R} and t>0, b>0, and \alpha >0 are constants. The nonlinear term f(v) is a given smooth function of v\in \mathbb{R}. More precisely,

where g(v)=O(v^2) for v\to 0.

This initial value problem was studied by [1, 2] when f satisfies

with p\ge 2, so that f^{\prime }(0)=0. Here C is independent of v, v_1, and v_2. [1] proved that there exists a global solution u\in C([0,\mathrm{\infty });H^2\cap L^1) to the problem (1.1), (1.2) under smallness condition on the initial data u_0\in W^{2,1}\cap H^2 and u_1\in L^2\cap L^1. In particular, they showed the decay estimates:

and

In addition to the above assumptions, suppose that the initial data u_0,u_1\in L_1^1, Takeda and Yoshikawa [2] established the following asymptotic profile of global solution:

where

and

The main purpose of our present paper is two-fold: first, we try to recover all the results about global existence and decay estimate of solution in Takeda and Yoshikawa [1] under some assumptions on the initial data (u_0(x),u_1(x)) and the nonlinear function f, which is much weaker than those needed in Takeda and Yoshikawa’s arguments. More precisely, the condition u_0(x)\in W^{2,1}\cap H^2, u_1(x)\in L^2\cap L^1 and (1.3) have been relaxed to u_0(x)\in H^{s+2}\cap L^1, u_1(x)\in H^s\cap L^1 (s\ge 0) and the nonlinear function f satisfies f^{\prime }(0)>-\alpha in this paper. Second, we show that the solution is asymptotic to a diffusion wave, given explicitly in terms of the solution of parabolic equation that is different from the one in [2]. For the details, we refer to Theorem 4.1. Moreover, under some additional assumptions on the initial data, we also prove that the convergence rates of our new asymptotic profile are better than that obtained by [2]. For details, we refer to Theorem 4.2.

The study of the global existence and asymptotic behavior of solutions to hyperbolic-type equations has a long history. We refer to [3, 4] for hyperbolic equations, [5–7] for the damped wave equation and [8–16] for various aspects of dissipation of the plate equation.

The paper is organized as follows. In Section 2, we study the decay property of the solution to the linear problem. Then, in Section 3, we prove the global existence and decay estimate of the solutions. Finally, we prove that the solution is asymptotic to a diffusion wave, which is given explicitly in terms of the solution of the parabolic equation in Section 4.

Notations We give some notations which are used in this paper. Let \mathcal{F}[u] denote the Fourier transform of u defined by

and we denote its inverse transform by {\mathcal{F}}^{-1}.

For 1\le p\le \mathrm{\infty }, L^p=L^p({\mathbb{R}}^n) denotes the usual Lebesgue space with the norm {\parallel \cdot \parallel }_{L^p}. The usual Sobolev space of s is defined by W^{s,p}={(I-{\partial }_x^2)}^{-\frac{s}{2}}{L}^p with the norm {\parallel f\parallel }_{W^{s,p}}={\parallel {(I-{\partial }_x^2)}^{\frac{s}{2}}f\parallel }_{L^p}.

Finally, in this paper, we denote every positive constant by the same symbol C or c which will not lead to any confusion. [\cdot ] is the Gauss symbol.

2.1 Solution formula

The aim of this subsection is to derive the solution formula for the problem (1.1), (1.2). We first investigate the linearized equation of (1.1):

with the initial data in (1.2), where \beta =\alpha +f^{\prime }(0)>0. We apply the Fourier transform to (2.1). This yields

The corresponding initial values are given as

The characteristic equation of (2.2) is

Let \lambda ={\lambda }_{\pm }(\xi ) be the corresponding eigenvalues of (2.4), we obtain

The solution to the problem (2.2), (2.3) in the Fourier space is then given explicitly in the form

where

and

We define G(x,t) and H(x,t) by

and

respectively, where {\mathcal{F}}^{-1} denotes the inverse Fourier transform. Then, applying {\mathcal{F}}^{-1} to (2.6), we obtain

By the Duhamel principle, we obtain the solution formula to (1.1), (1.2),

where g(v) is a smooth function; it satisfies g(v)=O(v^2).

2.2 Decay property

The aim of this subsection is to establish decay estimates of the solution operators G(t) and H(t) appearing in (2.9) and (2.10), respectively.

Lemma 2.1 The solution of the problem (2.2), (2.3) satisfies

for \xi \in \mathbb{R} and t\ge 0, where \rho (\xi )=\frac{{\xi }^2}{1+{\xi }^2}.

Proof Multiplying (2.2) by {\overline{\hat{u}}}_t and taking the real part yields

Multiplying (2.2) by \overline{\hat{u}} and taking the real part, we obtain

Multiplying both sides of (2.14) by 2 and summing up the resulting equation and (2.15) yields

where

and

A simple computation implies that

where

Note that

It follows from (2.17) that

Using (2.16) and (2.18), we get

Thus

which together with (2.17) proves the desired estimates (2.13). Then we have completed the proof of the lemma. □

Lemma 2.2 Let \hat{G}(\xi ,t) and \hat{H}(\xi ,t) be the fundamental solution of (2.1) in the Fourier space, which are given in (2.7) and (2.8), respectively. Then we have the estimates

and

for \xi \in \mathbb{R} and t\ge 0, where \rho (\xi )=\frac{{\xi }^2}{1+{\xi }^2}.

Proof Firstly, we investigate the problem (2.1), (1.2) with {\hat{u}}_0(\xi )=0, from (2.6), we obtain

Substituting the equalities into (3.1) with {\hat{u}}_0(\xi )=0, we get (2.19).

In what follows, we consider the problem (2.1), (1.2) with {\hat{u}}_1(\xi )=0; it follows from (2.6) that

Substituting the equalities into (2.13) with {\hat{u}}_1(\xi )=0, we get the desired estimate (2.20). The lemma is proved. □

Let

be the fundamental solution to u_t-\beta u_{xx}=0.

Lemma 2.3 Let \hat{G}(\xi ,t) and \hat{H}(\xi ,t) be the fundamental solution of (2.1) in the Fourier space, which are given in (2.6) and (2.7), respectively. Then there is a small positive number R_0 such that if |\xi |\le R_0 and t\ge 0, we have the following estimates:

and

Proof For sufficiently small ξ, using the Taylor formula, we get

We rewrite \hat{G}(\xi ,t) in (2.6) as

For sufficiently small ξ, from (2.24) and (2.25), we immediately obtain

For sufficiently small ξ, from (2.7) and (2.24), we immediately get

Thus we get (2.21). The other estimates are proved similarly and we omit the details. The proof of Lemma 2.3 is completed. □

Lemma 2.4 Let G(x,t) and H(x,t) be the fundamental solutions of (2.1), which are given in (2.9) and (2.10), respectively. Let 1\le p\le 2, and let k, j, and l be nonnegative integers. Then we have

for 0\le j\le k, where k+l-2\ge 0 in (2.26). Similarly, we have

for 0\le j\le k+2 in (2.28) and 0\le j\le k+4 in (2.29).

Proof We only prove (2.26). By the Plancherel theorem and (2.19), the Hausdorff-Young inequality, we obtain

For the term {\mathcal{A}}_1, letting \frac{1}{p^{\prime }}+\frac{1}{p}=1, we have

where we used the Hölder inequality with \frac{2}{p^{\prime }}+\frac{1}{q}=1 and the Hausdorff-Young inequality {\parallel \hat{v}\parallel }_{L^{p^{\prime }}}\le C{\parallel v\parallel }_{L^p} for v={\partial }_x^j\varphi. On the other hand, we can estimate the term {\mathcal{A}}_2 simply as

where k+l-2\ge 0.

Combining the above three inequalities yields (2.26). This completes the proof of Lemma 2.4. □

From Lemma 2.4, we immediately have the following corollary.

Corollary 2.1 Let G(x,t) and H(x,t) be the fundamental solution of (2.1), which are given in (2.8) and (2.9), respectively. Let 1\le p\le 2, and let k, j, and l be nonnegative integers. Then we have

for 0\le j\le k+1 and k+l-1\ge 0. Also we have

for 0\le j\le k+3.

Lemma 2.5 Let G(x,t) be the fundamental solution of (2.1), given in (2.8) and let G_0(x,t) be the fundamental solution of (2.1), given in (2.8). Let 1\le p\le 2, and let k, j, and l be nonnegative integers. Then we have

for 0\le j\le k+2. Similarly,

for 0\le j\le k+4.

Proof The proof of Lemma 2.5 is similar to the proof of Lemma 2.4. By employing (2.21) and (2.23), we can prove Lemma 2.5. We omit the details. □

The purpose of this section is to prove global existence and optimal decay estimate of solutions to the initial value problem (1.1), (1.2). We need the following lemma, which comes from [17] (see also [18]).

Lemma 3.1 Assume that f=f(v) is a smooth function. Suppose that f(v)=O({|v|}^{1+\theta }) (\theta \ge 1 is an integer) when |v|\le {\nu }_0. Then for integer m\ge 0, if v\in W^{m,q}({\mathbb{R}}^n)\cap L^p({\mathbb{R}}^n)\cap L^{\mathrm{\infty }}({\mathbb{R}}^n) and {\parallel v\parallel }_{L^{\mathrm{\infty }}}\le {\nu }_0, the following inequalities hold:

where \frac{1}{r}=\frac{1}{p}+\frac{1}{q}, 1\le p,q,r\le +\mathrm{\infty }.

Theorem 3.1 Let s\ge 0. Suppose that f(v) is a smooth function and f^{\prime }(0)>-\alpha. Assume that u_0\in H^{s+2}\cap L^1, u_1\in H^s\cap L^1. Put

Then there exists a positive constant {\delta }_1 such that if E_0\le {\delta }_0, and the initial value problem (1.1), (1.2) has a unique global solution u(x,t) satisfying

Moreover, the solution satisfies the decay estimate

and

where 0\le k\le s+2 in (3.2) and 0\le k\le s in (3.3).

Proof The existence and uniqueness of small solutions can be proved by the contraction mapping principle. Here we only show the decay estimates (3.2) and (3.3) for the solution u of (2.12) satisfying {\parallel u(t)\parallel }_{L^{\mathrm{\infty }}}\le M_0 with some M_0. To this end, we introduce the quantity

Here we note that

provided that s\ge 0. This follows from the Gagliardo-Nirenberg inequality, {\parallel u_x\parallel }_{L^{\mathrm{\infty }}}\le C{\parallel u_{xx}\parallel }_{L^2}^{\frac{1}{2}}{\parallel u_x\parallel }_{L^2}^{\frac{1}{2}} and the definition of X(t) in (3.4).

Applying {\partial }_x^k to (2.12) and taking the L^2 norm, we obtain

Firstly, we estimate I_1. We apply (2.26) with p=1, j=0 and l=0 (l=2 for k=0, l=1 for k=1). This yields

where {(k-2)}_{+}=max\{k-2,0\}. Similarly, applying (2.27) with p=1, j=0, and l=0 to the term I_2, we have

We estimate the nonlinear term J. We divide J into two parts and write J=J_1+J_2, where J_1 and J_2 are corresponding to the time intervals [0,t/2] and [t/2,t], respectively. For the term J_1, we apply (2.30) with p=1, j=0, and l=0. This yields

Here we see that {\parallel g(u_x)\parallel }_{L^1}\le C{\parallel u_x\parallel }_{L^2}^2 by Lemma 3.1. Thus we have {\parallel g(u_x)(\tau )\parallel }_{L^1}\le CX{(t)}^2{(1+\tau )}^{-\frac{3}{2}}. Therefore we can estimate the term J_{11} as

On the other hand, we have {\parallel {\partial }_x^{{(k-1)}_{+}}g(u_x)\parallel }_{L^2}\le C{\parallel u_x\parallel }_{L^{\mathrm{\infty }}}{\parallel {\partial }_x^{k}u\parallel }_{L^2} by Lemma 3.1. Therefore, using (3.5), we find that {\parallel {\partial }_x^{{(k-1)}_{+}}g(u_x)\parallel }_{L^2}\le CX{(t)}^2{(1+\tau )}^{-\frac{5}{4}-\frac{k}{2}}. Consequently, we can estimate the term J_{12} as

Finally, we estimate the term J_2 on the time interval [t/2,t]. Applying (2.30) with p=2, j=k, and l=0, and using {\parallel {\partial }_x^{{(k-1)}_{+}}g(u_x)(\tau )\parallel }_{L^2}\le CX{(t)}^2{(1+\tau )}^{-\frac{5}{4}-\frac{k}{2}}, we obtain

Thus we have shown that

Substituting all these estimates into (3.6), we obtain

for 0\le k\le s+2. Consequently, we have X(t)\le C{E}_0+CX{(t)}^2, from which we can deduce X(t)\le C{E}_0, provided that E_0 is suitably small. This proves the decay estimate (3.2).

In what follows, we prove the decay estimate (3.3) for the time derivative u_t. For this purpose we differentiate (2.12) with respect to t to obtain

Applying {\partial }_x^k to (3.11) and taking the L^2 norm, we have

where 0\le k\le s. For the term I_1^{\prime }, we apply (2.28) with p=1, j=0, and l=0 to get

Also, for the term I_2^{\prime }, applying (2.29) with p=1, j=0 and l=0, we have

To estimate the nonlinear term J^{\prime }, we rewrite J^{\prime }=J_1^{\prime }+J_2^{\prime }, where J_1^{\prime } and J_2^{\prime } correspond to the time intervals [0,t/2] and [t/2,t], respectively. For the term J_1^{\prime }, we apply (2.31) with p=1, j=0, and l=0. This yields

Since {\parallel g(u_x)(\tau )\parallel }_{L^1}\le CX{(t)}^2{(1+\tau )}^{-\frac{3}{2}} as before, we can estimate the term J_{11}^{\prime } as

Also, the term J_{12}^{\prime } is estimated similarly as before and we have J_{12}^{\prime }\le CX{(t)}^2{e}^{-ct}. Finally, we estimate the term J_2^{\prime } by applying (2.31) with p=2, j=k+2, and l=0 and obtain

where we used the estimate {\parallel {\partial }_x^{k+1}g(u_x)(\tau )\parallel }_{L^2}\le CX{(t)}^2{(1+\tau )}^{-\frac{9}{4}-\frac{k}{2}}. Consequently we have shown that

Substituting all these estimates together with the previous estimate X(t)\le C{E}_0 into (3.12), we arrive at the desired estimate (3.3) for 0\le k\le s. This completes the proof of Theorem 3.1. □

The above proof of Theorem 3.1 shows that the solution u to the integral equation (2.12) is asymptotic to the linear solution u_L given by the formula u_L(t):=G(t)\ast (u_0+u_1)+H(t)\ast u_0 in (2.11) as t\to \mathrm{\infty }. This result is stated as follows.

Corollary 3.1 Assume the same conditions of Theorem  3.1. Then the solution u of the problem (1.1), (1.2), which is constructed in Theorem  3.1, can be approximated by the solution u_L to the linearized problem (2.1), (1.2) as t\to \mathrm{\infty }. More precisely, we have the following asymptotic relations:

for 0\le k\le s+2 and 0\le k\le s, respectively, where u_L(t):=G(t)\ast (u_0+u_1)+H(t)\ast u_0 is the linear solution.

In this section, our aim is to establish an asymptotic profile to our global solution that is constructed in Theorem 4.1. In the previous section, we have shown that the solution u to the problem (1.1), (1.2) can be approximated by the linear solution u_L. In what follows, we shall derive a simpler asymptotic profile of the linear solution u_L.

Let v be the solution to the initial data problem

Then

gives a asymptotic profile of the linear solution u_L. In fact we have the following.

Lemma 4.1 Let s\ge 0. Assume that u_0,u_1\in H^{s+2}\cap L^1 and put E_0={\parallel (u_0,u_1)\parallel }_{H^{s+2}\cap L^1}. Let u_L be the linear solution and let v be defined by (4.2). Then we have

for 0\le k\le s+2.