This paper is devoted to a $p(x)$-Laplace equation with Dirichlet boundary. We obtain the existence of global solution to the problem by employing the method of potential wells. On the other hand, we show that the solution will blow up in finite time with $u_0 \not\equiv 0$ and nonpositive initial energy functional $J(u_0).$ By defining a positive function $F(t)$ and using the method of concavity we find an upper bound for the blow-up time.

In this paper, we study the solvability of a distribution-valued heat equation with nonlocal initial condition. Under proper assumption on parameters we get the explicit solution of the distribution-valued heat equation. As an application, we further consider the stabilization problem of heat equation with partial-delay in internal control. By the parameterization design of feedback controller, we show if the integral kernel functions are determined by the solution of the distribution heat equation with nonlocal initial value problem, then the closed-loop system can be transformed into a system which is called the target system of the exponential stability under the bounded linear transformation. By selecting different distribution-valued kernel functions, we give the inverse transformation. Hence the closed-loop system is equivalent to the target system.

In this note, we prove that the space of all admissible piecewise linear metrics parameterized by the square of length on a triangulated manifold is a convex cone. We further study Regge’s Einstein-Hilbert action and give a more reasonable definition of discrete Einstein metric than the former version. Finally, we introduce a discrete Ricci flow for three dimensional triangulated manifolds, which is closely related to the existence of discrete Einstein metrics.

In this paper, we study shrinking gradient Ricci solitons whose Ricci tensor has one eigenvalue of multiplicity at least $n−2.$ Firstly, we show that if the minimal eigenvalue of Ricci tensor has multiplicity at least $n−1$ at each point, then the soliton are Einstein. While on the shrinking gradient Ricci solitons whose maximal eigenvalue has multiplicity at least $n−1,$ the triviality are also true if we naturally require the positivity of Ricci tensor.
We further prove that if the maximal (or minimal) eigenvalue of Ricci tensor has multiplicity at least $n−2$ at each point , and in addition the sectional curvature is bounded from above, then the soliton are Einstein.

We introduce Besov spaces with variable smoothness and integrability by using the continuous version of Calderón reproducing formula. We show that our space is well-defined, i.e., independent of the choice of basis functions. We characterize these function spaces by so-called Peetre maximal functions and we obtain the Sobolev embeddings for these function spaces. We use these results to prove the atomic decomposition for these spaces.