The critical exponent of degenerate parabolic systems

The Cauchy problemut=Δuα +vp,vt=Δvβ +uq is studied, wherex εRN, 0 <t < ∞ and α,β,p andq, are positive exponents. It is proved that ifp,q ≥ 1 and 1 <pq < 1 + 2 max(p + β,q + α)/n then every nontrivial non-negative solution is not global in time; whereaspq > 1 + 2 max(p + β, q + α)/n then there exist both positive global solutions and non-global solutions. In addition, the decaying in time of solutions tout,=Δuα inRn × (0, ∞), an equation which occurs naturally in our study of above systems, is studied and solutions with the fastest decaying in time are constructed.

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