| In this paper, we study the large-time behavior of solutions of de-generate parabolic equations via coupled boundary sources. The main direction is to consider the existence and blow up of the solutions. We find out that the critical existence curve and the critical Fujita curve are same of this kind of system.In the first part, we introduce the background of the study, which is about the existence and blow up, of slow diffusion parabolic equations.In the second part, we have two main directions. Firstly, we show the system: where λ>p-1, λ2>q-1,p, q>2,α,β>0and u0(x),υ0(x) are continuous, nonnegative functions with compact support and satisfy some compatibility conditions. And we prove that the critical existence curve and the critical Fujita curve are same, which is αβ=(p-1)(q-1). Then,we extend the equatiions up to N-dimension,with N≥1,p,q>2,λ1≥p-N,λ2≥q-N,α,β≥0,B1(?)RN is the closed unit ball with the unit interior normall v to (?)B1,and u0(x),v0(x) are non-negative smooth functions with compact support and Satisfy some suitable compaatibility conditions.And we prove that the critical existence curve is the same with the Fujita curve of this N-dimensional equations. The curve is αβ=(p-1)(q-1). Secondly,we consider where λ1>p-1,λ2>q-1,m(p-1)>1,n(q-1)>1,α,β≥0, u0(x),v0(x)are continuous,nonnegative functions with compact support and satisfy some compatibility conditions.And we show that this system also has the property as the former one,that the critical existence curve and the critical Fujita curve are same,which is αβ=mn(p-1)(q-1), and we give a complete proof of the conclusion.We also extend this problem up to N-dimension:with N≥1,m(p-1)>1,n(q-1)>1,λ1≥p-N,λ2≥q-N,α,β卢≥0, B(?)RN is the closed unit ball with the unit interior normal v to (?)B1, and u0(x),v0(x)are non-negative smooth functions with compact support and satisfy some suitable compatibility conditions.And we prove that the critical existence curve and the critical Fujita curve are same,which is αβ=mn(p-1)(q-1). |
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