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Existence and nonexistence of global solutions for the generalized IMBq equation. (English) Zbl 0920.35005

We study the following initial-boundary value problem of the generalized IMBq equation on \(\overline Q_T= \overline\Omega\times [0,T]\) \((\Omega= (0,1), T>0)\): \[ u_{tt}- u_{xx}- u_{xxtt}= f(u)_{xx}, \]
\[ u(0,t)= u(1,t)= 0,\quad u(x,0)= u_0(x),\quad u_t(x, 0)= u_1(x), \] where \(u(x,t)\) denotes unknown function, \(f(s)\) is the given nonlinear function, \(u_0(x)\) and \(u_1(x)\) are the given initial value functions.
We prove the existence and uniqueness of the generalized local solution of the generalized global solution, the existence and uniqueness of the classical global solution, and blowing up of the solutions is discussed. In the final section, we consider the case \(f(u)= Ku^q\).

MSC:

35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35B40 Asymptotic behavior of solutions to PDEs
35Q35 PDEs in connection with fluid mechanics
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