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. We consider nonnegative solutions to the Cauchy problem for the quasilinear parabolic equations u_{t}=\triangle u^{m}+a\cdot\nabla u^{q}+u^{p} where m\geq 1 , p , q>1 , a\in R^{N} and a\neq 0 . In this paper we show: (a) if q>m-1 and \max\{m, q\}\leq p<\min\{m+2/N , m+2(q-m+ 1)/(N+1)\} or m+2/N\geq q\geq m+1/N and p=m+2/N , then all nontrivial solutions do not exist globally in time; (b) if p>m+2/N , then there are nontrivial global solutions. Further, in case (b) we study the asymptotic behavior of the global solutions. We also study the asymptotic behavior of the global solutions of u_{t}=\triangle u^{m}+a\cdot\nabla u^{q} .

Existence and nonexistence of global solutions to quasilinear parabolic equations with convection