Drops and Fingers in a Tempered Ginzburg-Landau set-up

We introduce gradient-tempered Ginzburg-Landau, GL, free energy functional J=∫(G(u_x)−W(u))dx, where W(u)=λ²(2u²−u⁴) is the bulk energy endowed with a stiffness coefficient a²(u) which vanishes both at the phases and on the interface: a²(u)=|u|^2γ|1−u²|^2β, [Formula presented]<β,γ≤1. It induces compact drops and interphase transitions within a finite domain. The assumed interface energy G=1+a²u_x²−1, tempers the divergence of gradients in the ultra-violet limit. Consequently, when λ², the bulk energy strength parameter crosses a critical value, depending on their amplitude, the forming drops may develop at their edges sharp jumps and turn into fingers across which the energy remains finite. In the Tempered Allen-Cahn, u_t=−[Formula presented]J equation, the resulting flux-saturating diffusion delays the resolution of initial discontinuities and if 1≤γ, it blocks excitation's spread beyond its initial span. A number of explicit spherical solutions is also presented.