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Although there is still no official policy paper or overall regional strategy towards Asia, the latest trend in Chinese diplomacy shows that China's strategy towards Asia is one of Harmonious Asia. The phrase connotes a sense of ‘togetherness’ that appeals strongly to the international society. To expatiate on the strategy, China proceeds along three tracks -economic cooperation, installing partnerships, and multilateral security. One could state that China's Asia strategy has been quite fruitful. However, there linger four future challenges to China's implementation of Harmonious Asia. That is, if the U.S. reinforces smart power; if Japan invents its own diplomacy; if China has difficulty harmonizing with regional states; and if China does not take sufficient responsibility. The possibility of voices calling for the need for a new system will rise once China becomes a responsible state and the regional states embrace China's subsequent leadership. Despite the potential and possibility, the reality of either a Sino-U.S. co-leadership structure or a Sino-U.S.-Japan tripartite concert will be unlikely to come to avail in the near future given the unremitting distrust between China and US/Japan on their strategic intentions vis-à-vis each other.

In this paper, we firstly discuss blow-up phenomena for nonlinear parabolic equations $$ u_{t}=\nabla \cdot \bigl[\rho (u)\nabla u \bigr]+f(x,t,u),\quad \text{in }\Omega \times \bigl(0,t^{*}\bigr), $$ u t = ∇ ⋅ [ ρ ( u ) ∇ u ] + f ( x , t , u ) , in  Ω × ( 0 , t ∗ ) , under mixed nonlinear boundary conditions $\frac{\partial u}{\partial n}+\theta (z)u=h(z,t,u)$ ∂ u ∂ n + θ ( z ) u = h ( z , t , u ) on $\Gamma _{1}\times (0,t^{*})$ Γ 1 × ( 0 , t ∗ ) and $u=0$ u = 0 on $\Gamma _{2}\times (0,t^{*})$ Γ 2 × ( 0 , t ∗ ) , where Ω is a bounded domain and $\Gamma _{1}$ Γ 1 and $\Gamma _{2}$ Γ 2 are disjoint subsets of a boundary ∂Ω. Here, f and h are real-valued $C^{1}$ C 1 -functions and ρ is a positive $C^{1}$ C 1 -function. To obtain the blow-up solutions, we introduce the following blow-up conditions: $$ (C_{\rho})\,:\, \begin{aligned} &(2+\epsilon ) \int _{0}^{u}\rho (w)f(x,t,w)\,dw\leq u\rho (u)f(x,t,u)+ \beta _{1}u^{2}+\gamma _{1}, \\ &(2+\epsilon ) \int _{0}^{u}\rho ^{2}(w)h(z,t,w)\,dw \leq u\rho ^{2}(u)h(z,t,u)+ \beta _{2}u^{2}+ \gamma _{2}, \end{aligned} $$ ( C ρ ) : ( 2 + ϵ ) ∫ 0 u ρ ( w ) f ( x , t , w ) d w ≤ u ρ ( u ) f ( x , t , u ) + β 1 u 2 + γ 1 , ( 2 + ϵ ) ∫ 0 u ρ 2 ( w ) h ( z , t , w ) d w ≤ u ρ 2 ( u ) h ( z , t , u ) + β 2 u 2 + γ 2 , for $x\in \Omega $ x ∈ Ω , $z\in \partial \Omega $ z ∈ ∂ Ω , $t>0$ t > 0 , and $u\in \mathbb{R}$ u ∈ R for some constants ϵ, $\beta _{1}$ β 1 , $\beta _{2}$ β 2 , $\gamma _{1}$ γ 1 , and $\gamma _{2}$ γ 2 satisfying $$ \epsilon >0,\quad \beta _{1}+\frac{\lambda _{R}+1}{\lambda _{S}}\beta _{2} \leq \frac{\rho _{m}^{2}\lambda _{R}}{2}\epsilon \quad \text{and}\quad 0 \leq \beta _{2}\leq \frac{\rho _{m}^{2}\lambda _{S}}{2}\epsilon , $$ ϵ > 0 , β 1 + λ R + 1 λ S β 2 ≤ ρ m 2 λ R 2 ϵ and 0 ≤ β 2 ≤ ρ m 2 λ S 2 ϵ , where $\rho _{m}:=\inf_{s>0}\rho (s)$ ρ m : = inf s > 0 ρ ( s ) , $\lambda _{R}$ λ R is the first Robin eigenvalue and $\lambda _{S}$ λ S is the first Steklov eigenvalue. Lastly, we discuss blow-up solutions for nonlinear parabolic systems.

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