Existence and uniform decay rates of solutions to a degenerate system with memory conditions at the boundary

In this article we study the degenerate system (@r"1,@r"2>=0) subject to memory conditions on the boundary given [email protected]"1(x)u"t"[email protected][email protected](u-v)[email protected]]0,+~[,@r"2(x)v"t"[email protected]@a(u-v)[email protected]]0,+~[,[email protected]"0,[email protected]!"0^tg"1(t-s)@[email protected][email protected](s)[email protected]"1x]0,+~[,[email protected]"0,[email protected]!"0^tg"2(t-s)@[email protected][email protected](s)[email protected]"1x]0,+~[,(u(0),v(0))=(u^0,v^0)(@r"1u"t(0),@r"2v"t(0))=(@r"1u^1,@r"2v^1)[email protected],where @W is a bounded region in R^n whose boundary is partitioned into disjoint sets @C"0, @C"1. We prove that the dissipations given by the memory terms are strong enough to guarantee exponential (or polynomial) decay provided the relaxation functions also decay exponentially (or polynomially) and with the same rate of decay.

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