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Oscillation criteria for impulsive parabolic differential equations with delay. (English) Zbl 1160.35429

The authors investigate a class of nonlinear impulsive parabolic systems with delay, \[ u_t=\sum\limits_{i=1}^{n}a_i(t)\partial^2u/\partial x_i^2-g(t,x)h(u(t-r,x)), \;\;\;\;t\neq t_k \] \(\triangle u=I(t,x,u)\), \(t=t_k\) (\(k=1,2,3,\dots\)), where (i) \(0<t_1<t_2<\cdots <t_k<\cdots \) and \(\lim_{k\to\infty }t_k=+\infty \), (ii) \(\triangle u| _{t=t_k}=u(t_k^+,x)-u(t_k^-,x)\), (iii) the unknown function \(u(t,x)\) is defined in \(R_+\times \Omega \) (\(\Omega \) is a bounded domain in \(R^n\) with a smooth boundary), (iv) \(r>0\), \(a_i\in PC[{}R_+,R_+]{}\) (i=1,2,3,…,n), where \(PC\) denotes the class of functions which are piecewise continuous in \(t\) with discontinuities of the first kind only at \(t=t_k\) (\(k=1,2,\dots\)), and left continuous in \(t=t_k\), \(g\in PC[{}R_+\times \bar\Omega ,R_+]{}\), \(h\in C[{}R,R]{}\), (v) \(I:R_+\times\bar\Omega\times R\to R\). Two conditions are assumed to be satisfied, (H1) \(h(u)\) is a positive and convex function in the segment \(0,+\infty \), (H2) For any function \(u\in PC[{}R_+\times \bar\Omega ,R_+]{}\) and constants \(\alpha_k\) satisfy certain integral inequalities. Then under these conditions the authors prove that each nonzero solution of the considered problem is oscillatory in some domain. Other oscillation criteria are established for the system subject to two different boundary conditions by employing Gauss’ divergence theorem and certain impulsive delay differential inequalities.

MSC:

35K40 Second-order parabolic systems
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35R10 Partial functional-differential equations
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References:

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