Stability results for hyperbolic and parabolic equations. (English) Zbl 0872.35012
Our aim is to present some stability results for hyperbolic and parabolic PDEs and to review some recent results in this field. Essentially, we consider the following PDEs:
\[
L_1u_{tt}+ L_2u_t+ L_3u= f(u),\tag{1}
\]
\[ u_{tt}= g(u_x)u_{xx}+ u_{xtx},\tag{2} \]
\[ u_t= [F(x,u(x,t)]_{xx},\tag{3} \] where \(u:(0,1)\times\mathbb{R}_+\to \mathbb{R}\), \(f\), \(g\) and \(F\) are suitable real functionals such that \(f(0)= F(x,0)= 0\); \(L_i\) \((i=1,2,3)\) are linear (spatial) selfadjoint operators, defined on a dense subset \(S\) of a real Hilbert space. To equations (1)–(3) we append the homogeneous boundary conditions \[ u(0,t)= u(1,t)=0 \] and suitable initial data. We recall the fundamental role played by the Lyapunov function in the stability theory of dynamical systems and the essential steps of the Lyapunov direct method. Further, we consider the hyperbolic equations (1)–(2) introducing for them suitable Lyapunov functions. Successively, we apply one of the introduced Lyapunov functions to a generalized Maxwell-Cattaneo equation. Finally, we consider equation (3) and establish some conditions ensuring that the solutions to the same boundary data are asymptotically the same, regardless of the initial value.
\[ u_{tt}= g(u_x)u_{xx}+ u_{xtx},\tag{2} \]
\[ u_t= [F(x,u(x,t)]_{xx},\tag{3} \] where \(u:(0,1)\times\mathbb{R}_+\to \mathbb{R}\), \(f\), \(g\) and \(F\) are suitable real functionals such that \(f(0)= F(x,0)= 0\); \(L_i\) \((i=1,2,3)\) are linear (spatial) selfadjoint operators, defined on a dense subset \(S\) of a real Hilbert space. To equations (1)–(3) we append the homogeneous boundary conditions \[ u(0,t)= u(1,t)=0 \] and suitable initial data. We recall the fundamental role played by the Lyapunov function in the stability theory of dynamical systems and the essential steps of the Lyapunov direct method. Further, we consider the hyperbolic equations (1)–(2) introducing for them suitable Lyapunov functions. Successively, we apply one of the introduced Lyapunov functions to a generalized Maxwell-Cattaneo equation. Finally, we consider equation (3) and establish some conditions ensuring that the solutions to the same boundary data are asymptotically the same, regardless of the initial value.
MSC:
| 35B35 | Stability in context of PDEs |
| 35G10 | Initial value problems for linear higher-order PDEs |
| 35K25 | Higher-order parabolic equations |
| 35L70 | Second-order nonlinear hyperbolic equations |
| 35K65 | Degenerate parabolic equations |
Keywords:
homogeneous boundary conditions; Lyapunov function; Lyapunov direct method; generalized Maxwell-Cattaneo equationReferences:
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