Existence,uniqueness, and nonexistence of solutions to nonlinear diffusion equations with p ( x , t ) $p(x,t)$ -Laplacian operator

Abstract

The aim of this paper is to deal with the existence and nonexistence of weak solutions to the initial and boundary value problem for u t = div(|∇u|. By constructing suitable function spaces and applying the method of Galerkin's approximation as well as weak convergence techniques, the authors prove the existence of local solutions. Furthermore, we choose a suitable test-function, make integral estimates, and apply Gronwall's inequality to prove the uniqueness of weak solutions. At the end of this paper, the authors construct a suitable energy functional, obtain a new energy inequality, and apply a convex method to prove the nonexistence of solutions. Especially, it is worth pointing out that the results are obtained with the assumption that p t (x, t) is only negative and integrable, which is weaker than most of the other papers required.

Power law Starobinsky model of inflation from no-scale SUGRA

“…(16) into the potential term of Eq. (14) and simplifying, we get the effective scalar potential in the Einstein frame as …”

“…In this section we will show that generalized non-minimally coupled inflation models ξ a R b [11] with the quantum corrected 4 -potential [12][13][14] can be reduced to the power law Starobinsky form. We consider the generalized non-minimal coupling ξ a R b and the quantum correction to quartic scalar potential 4(1+γ ) into the action…”

“…A general scalar-curvature coupled ξφ a R b model was studied in [11]. The quantum correction on φ 4 -potential in Jordan frame was studied in [12][13][14] where the equivalence of the ξφ 2 R + λφ 4(1+γ ) model with 1 M 2 R β model was shown. The generalized Starobinsky model with R p correction has been studied in Refs.…”

Axion landscape and natural inflation

“…9 The tadpole condition is another important issue for implementing the KNP mechanism in the string theory, which requires a number of (brane) charges through the alignment of wrapping numbers of branes or gauge fluxes on branes in the extra dimensions, while the net charge of branes should be vanishing in the compact extra dimension. 10 In this respect, it is interesting to consider a possibility that the inflaton consists of RR two-form axions in type IIB model [42,44,45,47] and fluxed seven branes, which produce non-perturbative inflaton potential, are wrapping on a curved extra dimension with orientifolds. In this case, a tight consistency condition on three brane charges can be relaxed by the curvature corrections due to the seven branes [79,80].…”

“…The KNP mechanism has attracted much attention especially after the BICEP2 result and it has been studied from various aspects [18,[37][38][39][40][41][42][43][44][45][46][47]. The original KNP mechanism [36] relied upon two axions with sub-Planckian decay constants, and a relatively large hierarchy in the anomaly coefficients was required for successful inflation.…”