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
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.
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