Blow-up of solutions for improved Boussinesq type equation
The authors investigate a one-dimensional type of Boussinesq equations, i.e. \[ u_{tt}- u_{xx}- u_{xxtt}= \sigma(u)_{xx},\;x\in (0,1),\;t\in (0,\infty),\tag{1} \] \[ u_x(0, t)= 0,\;u_x(1,t)= 0,\;t> 0,\;u(x,0)= \varphi(x),\;u_t(x, 0)= \psi(x). \] The authors are also interested in the case where \(\sigma(u)\) has the particular form \[ \sigma(u)= au^p,\quad p> 1,\quad a\neq 0.\tag{2} \] There are two types of results. The first, Theorem 2.1, asserts the existence and uniqueness of a generalized solution \(u\in W^{2,\infty}([0, T_m),A)\) on a maximal interval of existence \([0,T_m)\) \((T_m\leq\infty)\) where \(A\) is a suitably defined Sobolev space explained in terms of Fourier series; the notion of generalized solution used is introduced in the course of the proof. The other type of theorems are blow-up results which apply to the case where \(\sigma(u)\) is of the form (2). Thus with \(\varphi\), \(\psi\) as in (1), and with \[ \varphi_k= 2\int^1_0 \varphi(x)\cos\pi kx dx,\quad \psi_k= 2 \int^1_0 \psi(x)\cos\pi kx dx,\tag{3} \] Theorem 2.2 states that blow-up takes place if \(\sigma(u)\) is of the form (2) and if in addition: (a) \(\varphi_k,\psi_k\leq 0\) for \(k\geq 0\), (b) \(\varphi_0\leq -2(|a|p)^{-1/(p-1)}\), (c) \(a< 0\), \(p\geq 3\) is odd and \(\psi_1< 0\). Further, but more complicated conditions for blow-up are provided by Theorem 2.3. The proofs are via a series of rather involved lemmas.
scientific article; zbMATH DE number 1915713
scientific article