| Budget Amount *help |
¥3,310,000 (Direct Cost: ¥3,100,000、Indirect Cost: ¥210,000)
Fiscal Year 2007: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2006: ¥700,000 (Direct Cost: ¥700,000)
Fiscal Year 2005: ¥1,700,000 (Direct Cost: ¥1,700,000)
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| Research Abstract |
In our project, we study the asymptotic behavior of nonnegative solutions of the Dirichlet problem(Ω is bounded) or the Cauchy problem(Ω = R^N) for a quasilinear parabolic equation with a heat source : u_t-Δu^m= F in(x, t)∈ Ω ×(0, T), where m ≧1, and F =f(u)(a usual heat source) or F = f(u(x_0(t), t))(x_0(t)∈Ω)(localized reaction).Furthermore, we assume that f satisfies some blow-up condition. This equation represents various phenomena and gives interesting various problems. We have obtained the next three results for these problems.
(i) When m=1, Ω is a bounded domain and F=f(u(x_0(t), t)), we showed that the boundedness of global solutions is determined by the asymptotic behavior of x_0(t)as t→∞.This result is a part of our result on the classification of all solutions. However, when m> 1, we do not have good results for this problem, since we do not know whether or not the uniqueness of solutions holds.
(ii)When m ≧1, Ω= R^N and F=u^P , we studied the precise behavior of solutions which blow up at space infinity. In particular, we introduced "blow-up solution with the least blow-up time" and showed that such a solution blows up at space infinity. We give a necessary and sufficient condition for a solution to be a blow-up solution with the least blow-up time. We also give a necessary and sufficient condition for a blow-up solution with the least blow-up time to blow up in a direction ψ.
(iii)When m>1, Ω= R^N and F = u^P , we studied under what condition the solution blows up in finite time, and got new results.
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