Finger-like smoldering patterns are observed in experiments under microgravity. For theoretical understanding of such pattern phenomena, a model of reaction-diffusion system has been proposed. In this paper, we prove the existence and uniqueness of a solution for this reaction-diffusion system. We also consider a large-time behavior of solutions.

Our aim in this paper is to deal with continuity properties for logarithmic potentials of functions in Morrey spaces of variable exponent. Our exponent approaches 1 on some part of the domain, and hence continuity properties depend on the shape of that part and the speed of the exponent approaching 1.

We construct a sequentially compact space of patches. By using this construction, we analyze certain symmetries of tilings obtained by substitution rules.

In this short note we discuss the influence of a time-periodic dissipation term on a-priori estimates for solutions to dissipative wave equations. The approach is based on a diagonalisation argument for high frequencies and results from spectral theory of periodic differential equations / Floquet theory for bounded frequencies.

Necessary and sufficient conditions are given for a satellite knot to be fibered. Any knot $\tilde k$ embeds in an unknotted solid torus $\tilde V$ with arbitrary winding number in such a way that no satellite knot with pattern $(\tilde V, \tilde k)$ is fibered. In particular, there exist nonfibered satellite knots with fibered pattern and companion knots and nonzero winding number.

Our aim is to give an elementary proof of Sobolev embeddings for Riesz potentials of functions in Morrey spaces $L^{1,\nu,\beta}(G)$, as an extension of Serrin. We are mainly concerned with Trudinger's type exponential integrability for Riesz potentials.

We define elementary automorphisms of the $n$-dimentional vector group over an algebraically closed field of positive characteristic and show that they generate the automorphism group of the vector group. We also give a necessary and sufficient computational condition for a $d$-tuple of $p$-polynomials to be a component of an $n$-tuple of $p$-polynomials defining an automorphism of the vector group.

We show that there exist infinitely many non-invertible, hyperbolic knots that admit toroidal Dehn surgery of hitting number four. The resulting toroidal manifold contains a unique incompressible torus meeting the core of the attached solid torus in four points, but no incompressible torus meeting it less than four points.

A superharmonic function $u$ on a parabolic Riemannian manifold $M$ is shown to admit the Riesz decomposition $u=h+(1/c_{d})\int_{M}e(\cdot,y)d\mu(y)$ on $M$ into the harmonic function $h$ on $M$ and the Evans potential of an Evans kernel $e(x,y)$ on $M$ and of the Borel measure $\mu:=-\Delta u\geqq 0$ on $M$ multiplied by a certain constant $1/c_{d}$ if and only if $m(t^{2},u)-2m(t,u)={\cal O}(1)\ (t\rightarrow+\infty)$, where $m(t,u)$ is the spherical mean over the sphere of radius $t$ all induced by the above chosen Evans kernel $e(x,y)$ on $M$.

The parabolic Bergman space is the set of all $L^p$-solutions of the parabolic operator $L^{(\alpha)}$. In this paper, we study fractional calculus on parabolic Bergman spaces. In particular, we investigate properties of fractional derivatives of the fundamental solution of the parabolic operator. We show the reproducing property of fractional derivatives of the fundamental solution.