NONLINEAR INSTABILITY OF SOLUTIONS IN PARABOLIC AND HYPERBOLIC DIFFUSION

作者：Pankavich Stephen^*; Radu Petronela

来源：Evolution Equations and Control Theory, 2013, 2(2): 403-422.

DOI：10.3934/eect.2013.2.403

• Evolution equations
• sign-changing damping
• instability
• variable coefficients
• steady states

We consider semilinear evolution equations of the form a(t)partial derivative(tt)u + b(t)partial derivative(t)u + Lu = f (x, u) and b(t)partial derivative(t)u + Lu = f (x, u), with possibly unbounded a(t) and possibly sign-changing damping coefficient b(t), and determine precise conditions for which linear instability of the steady state solutions implies nonlinear instability. More specifically, we prove that linear instability with an eigenfunction of fixed sign gives rise to nonlinear instability by either exponential growth or finite-time blow-up. We then discuss a few examples to which our main theorem is immediately applicable, including evolution equations with supercritical and exponential nonlinearities.

• 出版日期2013-6
• 单位中国人民解放军海军大连舰艇学院