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MATHEMATICS
V. A. TRENOGIN
EXISTENCE AND ASYMPTOTICS OF SOLUTIONS OF THE “SOLITARY WAVE” TYPE FOR DIFFERENTIAL EQUATIONS IN A BANACH SPACE
(Presented by Academician S. L. Sobolev on 20 I 1964)
The problem considered in this note describes a phenomenon well known in mechanics under the name “solitary wave” and first studied by M. A. Lavrent’ev \((^1)\) (see also \((^{2,3})\)). Recently A. M. Ter-Krikorov and the author showed \((^4)\) that this phenomenon occurs for a certain class of quasilinear elliptic equations in a rectilinear unbounded strip. Below, the validity of analogous results is established for a second-order differential equation in a Banach space. Our investigation makes essential use of the theory of semigroups.
Lemma 1. Let an operator \(B\) act in a Banach space \(E\), where:
1) \(B\) is a closed linear unbounded operator with domain dense in \(E\);
2) \(-B\) is the infinitesimal generator of a strongly continuous semigroup \(\exp(-\xi B)\), \(\xi \geqslant 0\), such that
\[ \|\exp\{-\xi B\}\| \leqslant \exp\{-\xi\beta\}, \qquad \beta=\operatorname{const}>0. \]
Define \(B^{1/2}\) to be a closed linear unbounded operator with domain dense in \(E\), as the inverse of the operator (see \((^5)\))
\[ B^{-1/2}=\frac{1}{\sqrt{\pi}}\int_{0}^{+\infty}\exp\{-\eta B\}\eta^{-1/2}\,d\eta . \]
Then the operator \(-B^{1/2}\) is the infinitesimal generator of the strongly continuous semigroup \(\exp\{-\xi B^{1/2}\}\), \(\xi \geqslant 0\), and moreover
\[ \|\exp\{-\xi B^{1/2}\}\| \leqslant \exp\{-\xi\sqrt{\beta}\}. \]
Proof. A related result is available in \((^6)\). Our proof is based on the following integral representation, valid for \(\lambda \in (-\sqrt{\beta},+\infty)\):
\[ (B^{1/2}+\lambda I)^{-1} = \frac{1}{\sqrt{\pi}} \int_{0}^{+\infty} \exp\{-\eta B\}\psi(\lambda\sqrt{\eta})\eta^{-1/2}\,d\eta, \]
where
\[ \psi(x)=1-2xe^{x^2}\int_{x}^{+\infty}e^{-t^2}\,dt>0. \]
From this representation one derives an estimate for the norm of the operator \((B^{1/2}+\lambda I)^{-1}\), which makes it possible to apply the Hille–Yosida–Phillips theorem \((^7)\) and obtain the assertion of the lemma.
We now introduce, for \(\delta>0\), \(E_\delta\)—the Banach space of abstract even functions \(f(\xi)\), continuous on \((-\infty,+\infty)\), with norm
\[ |f(\xi)|_\delta=\sup_{(-\infty,+\infty)} \|e^{\delta|\xi|}f(\xi)\|. \]
Lemma 2. Suppose that the conditions of Lemma 1 are fulfilled for the operator \(B\), and suppose \(h(\xi)\in E_\delta,\ h'(\xi)\in E_\delta\), where \(0<\delta<\sqrt{\bar\beta}\). Then the boundary-value problem
\[ -\frac{d^2 z}{d\xi^2}+Bz=h(\xi),\qquad \lim_{\xi\to\pm\infty} z=0 \]
has a solution \(z=z(\xi)\), \(z^{(k)}(\xi)\in E_\delta,\ k=0,1,2\). This solution is given by the explicit formula
\[ z(\xi)=\frac12\int_{-\infty}^{+\infty} \exp\{-|\xi-\eta|B^{1/2}\}B^{-1/2}h(\eta)\,d\eta . \]
The validity of Lemma 2 is verified by direct computations. We now pass to the problem of a solitary wave; namely, we consider in the Banach space \(E\) the following nonlinear boundary-value problem:
\[ -\frac{d^2 y}{d\eta^2}+Ay=F(\lambda,y),\qquad -\infty<\eta<+\infty,\qquad \lim_{\eta\to\pm\infty}y(\eta)=0. \tag{1} \]
Here \(\lambda\) is a real parameter; \(A\) is a closed linear unbounded operator with dense domain of definition in \(E\); \(F(\lambda,y)\) is a nonlinear operator acting in \(E\), analytic in the Fréchet sense with respect to \(\lambda,y\) in some neighborhood of the point \(y=0\) for all \(\lambda\), and \(F(\lambda,0)=0\). Problem (1) always has the trivial solution. Our aim is to give conditions sufficient for a small nontrivial solution to branch off from the trivial solution for some \(\lambda=\lambda_0\).
Suppose there exists \(\lambda_0\) such that the operator \(B=A-\partial F(\lambda_0,0)/\partial y\) satisfies the following conditions:
1) Zero is a simple isolated eigenvalue of the operator \(B\), with corresponding null element \(\varphi\).
2) For solvability of the equation \(By=h\) it is necessary and sufficient that \(\psi(h)=0\), where \(\psi\) is some linear functional in \(E\); moreover \(\varphi\) and \(\psi\) can be normalized so that \(\psi(\varphi)=-1\).
3) Decompose \(E\) into the direct sum \(E=E^1\dot{+}E^{\infty-1}\), where \(E^1\) is the null subspace of the operator \(B\), and \(E^{\infty-1}\) is its range. In \(E^{\infty-1}\) the operator \(B\) is invertible. We require that, for \(B\) considered on \(E^{\infty-1}\), the conditions of Lemma 1 be fulfilled. Put now \(\lambda=\lambda_0+\varepsilon\) and write \(F(\lambda,y)\) in the form
\[ F(\lambda_0+\varepsilon,y)=F_0y+\sum_{j=2}^{\infty}F_{0j}y^j+ \sum_{i=1}^{\infty}\sum_{j=1}^{\infty}F_{ij}\varepsilon^i y^j . \]
The subsequent exposition uses the definition of a generalized Jordan chain, which generalizes the previously given definitions (see \((^4,^8,^9)\)).
Let the operator \(\Phi(y)\) be analytic in the Fréchet sense in some neighborhood of zero, with
\(\Phi^{(i)}(0)=0,\ i=0,1,\ldots,m-1\), but \(\Phi^{(m)}(0)\ne0\). Let \(\Phi_i(y_1,y_2,\ldots,y_i)\) be the coefficients in the formal expansion
\[ \Phi\left(\sum_{k=1}^{\infty} y_k\varepsilon^k\right) = \sum_{i=1}^{\infty}\Phi_i(y_1,y_2,\ldots,y_i)\varepsilon^{m+i-1}. \]
Definition. We shall say that the operator \(B\) has, relative to the operator \(\Phi(y)\), a Jordan chain of length \(\rho\), if there exist \(\rho\) linearly independent elements
\(\varphi_1, \varphi_2, \ldots, \varphi_p\), satisfying the relations \(B\varphi_1=0\), \(B\varphi_k=\Phi_{k-1}(\varphi_1,\ldots,\varphi_{k-1})\), \(k=2,\ldots,p\), and moreover
\[ \psi\bigl(\Phi_p(\varphi_1,\ldots,\varphi_p)\bigr)\ne 0. \]
Theorem 1. Suppose that the operator \(B\) has, relative to the operator
\[ \Phi(y)=\sum_{j=2}^{\infty} F_{0j}y^j, \]
a Jordan chain of length \(p\), and suppose that \(\alpha=\psi(F_{11}\varphi)>0\). Put \(a_p=\psi\bigl(\Phi_p(\varphi_1,\ldots,\varphi_p)\bigr)\).
Then, if \(p(m-1)\) is odd, for all \(\lambda\) sufficiently close to \(\lambda_0\), and if \(p(m-1)\) is even, for all \(\lambda\) sufficiently close to \(\lambda_0\) and satisfying the inequality \(a_p(\lambda_0-\lambda)>0\), there exists a nontrivial solution of problem (1) in the space \(E_\delta\), where \(0<\delta<\min(\sqrt{\beta},\,2(m-1)\sqrt{\alpha})\). The following asymptotics hold \((\lambda>\lambda_0,\ N>p)\):
\[ y(\eta,\varepsilon)=\sum_{i=1}^{N} y_i\bigl(\eta\sqrt{\lambda-\lambda_0}\bigr) (\lambda-\lambda_0)^{[1+(i-1)(m-1)]/p(m-1)} + \]
\[ +\,O\bigl((\lambda-\lambda_0)^{[1+N(m-1)]/p(m-1)}\bigr). \]
Remark. Together with \(y(\eta)\), \(y(\eta+c)\) will be a solution of problem (1) for any \(c\), since (1) is invariant with respect to shifts in \(\eta\). The asymptotics is constructed in the same way as in (4); its principal term has the form
\[ \left[\frac{\alpha(\lambda_0-\lambda)}{a_p}\right]^{1/p(m-1)} \left[\operatorname{ch}\frac{1}{2}p(m-1)\sqrt{\alpha|\lambda_0-\lambda|}\,\eta\right]^{-2/p(m-1)} \varphi . \]
For the proof of the existence of a solution, Lemma 2 is used.
The solitary wave found in Theorem 1 has exponential decay at infinity. We give conditions under which the decay is of power type.
Let \(L_{ij}\) be the coefficients of the branching equation composed for finding small solutions of the equation \(Ay=F(\lambda_0+\varepsilon,y)\) (see \((10,11)\)). Introduce \(E_*\), the Banach space of abstract even functions \(f(\xi)\), continuous on \((-\infty,+\infty)\), with norm
\[ \lvert f(\xi)\rvert_*=\sup_{(-\infty,+\infty)} \|f(\xi)\omega(\xi)\|, \]
where
\[ \omega(\xi)=(1+\xi^2)^{3/2}\ln(e+|\xi|). \]
Theorem 2. Suppose \(L_{11}=L_{12}=L_{20}=0\), \(L_{21}\ne0\), \(L_{30}<0\). Then, for all \(\lambda\) sufficiently close to \(\lambda_0\), there exists a nontrivial solution of problem (1) in the space \(E_*\). The asymptotics holds
\[ y(\eta,\varepsilon)=\sum_{i=1}^{N} y_i\bigl(\eta(\lambda-\lambda_0)\bigr)(\lambda-\lambda_0)^i +O\bigl((\lambda-\lambda_0)^{N+1}\bigr). \]
The scheme of the proof is the same. We note only that in Lemma 2 the space \(E_\delta\) can be replaced by the space \(E_*\), and lemmas analogous to Lemmas 1 and 2 from (4) can be proved. The principal term of the asymptotics here will be
\[ \frac{6(\lambda-\lambda_0)}{L_{21}} \left[\eta^2(\lambda-\lambda_0)^2-\frac{9}{2}L_{30}\right]^{-1} \varphi . \]
Example. \(E=L_q(\Omega)\), where \(\Omega\) is a simply connected bounded domain in \(R^n\) with boundary \(\Gamma\),
\[ Ay\equiv-\sum_{i,j=1}^{n} a_{ij}(\xi)\frac{\partial^2 y}{\partial \xi_i\partial \xi_j} +\sum_{i=1}^{n} a_i(\xi)\frac{\partial y}{\partial \xi_i} +a(\xi)y, \]
\(a_{ij}, a_i, a\) are sufficiently smooth, \(\xi=(\xi_1,\ldots,\xi_n)\), and
\[ \sum_{i,j=1}^{n} a_{ij}(\xi)\gamma_i\gamma_j \geq k \sum_{i=1}^{n}\gamma_i^2,\qquad k=\text{const}>0. \]
The domain of definition of \(A\) consists of functions belonging to \(W_q^{(2)}(\Omega)\) for which \(y|_\Gamma=0\). Let
\[ F(\lambda,y)=\sum_{i=1}^{n} F_i(\lambda,\xi)y^i, \]
where \(F_i(\lambda,\xi)\) are sufficiently smooth. In the infinite cylinder \(\Omega\times(-\infty,+\infty)\) consider the elliptic equation
\[ \frac{\partial^2 y}{\partial \eta^2} +\sum_{i,j=1}^{n} a_{ij}(\xi)\frac{\partial^2 y}{\partial \xi_i\partial \xi_j} -\sum_{i=1}^{n} a_i(\xi)\frac{\partial y}{\partial \xi_i} -a(\xi)y+F(\lambda,\xi,y)=0 \]
with boundary conditions
\[ y\big|_{\Gamma\times(-\infty,+\infty)}=0,\qquad \lim_{\eta\to\pm\infty} y=0. \]
For this problem Theorems 1 and 2 can be rephrased. The conditions of Lemma 1 are verified in \((^{12})\). One can also indicate conditions under which the solution found will be classical.
Moscow Institute of Physics and Technology
Received
15 I 1964
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