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In this paper, we discuss the conditions under which blow-up occurs for the solutions of reaction-diffusion equations on networks. The analysis of this class of problems includes the existence of blow-up in finite time and the determination of the blow-up time and the corresponding blow-up rate. In addition, when the solution blows up, we give estimates for the blow-up time and also provide the blow-up rate. Finally, we show some numerical illustrations which describe the main results. [ABSTRACT FROM AUTHOR]

In this paper, we define the discrete p-Schrodinger operators on finite networks and discuss the existence of the Dirichlet eigenvalues and their eigenfunctions for the operators. We also provide various equivalent conditions for the existence of positive solutions for Dirichlet boundary value problems for the operators. [ABSTRACT FROM AUTHOR]

We develop a comparison principle for the discrete p-Laplacian Δp,ω as follows: Let S be a connected graph. Assume that f1 and f2 are functions on S satisfying[image omitted] If f1≢f2 on S, then f1 < f2 in S. Moreover, the comparison principles for problems with various boundary data are introduced. Especially, for a problem without boundary data, we find that a supersolution (or subsolution) is equivalent to a solution of an equation. [ABSTRACT FROM AUTHOR]

We consider the general solution of quartic functional equations and prove the Hyers-Ulam- Rassias stability. Moreover, using the pullbacks and the heat kernels we reformulate and prove the stability results of quartic functional equations in the spaces of tempered distributions and Fourier hyperfunctions. [ABSTRACT FROM AUTHOR]

Let E(x, ξ, t) be the Mehler kernel and let U(x, t) be the solution, with some appropriate growth estimate (*), of the Hermite heat equation in R×(0, T). Then there exists a unique Fourier hyperfunction u such that U(x, t)=〈 u(ξ), E(x, ξ, t)〉. Conversely for any Fourier hyperfunction u, U(x, t):=〈 u(ξ), E(x, ξ, t)〉 is the smooth solution of the Hermite heat equation in R×(0, T) satisfying the growth estimate (*). [ABSTRACT FROM AUTHOR]

Abstract Meisters and Peterson gave an equivalent condition under which the multisensor deconvolution problem has a solution when there are two convolvers, each the characteristic function of an interval. In this article we find additional conditions under which the deconvolution problem for multiple characteristic functions is solvable. We extend the result to the space of Gevrey distributions and prove that every linear operator S, fromthe space of Gevrey functions with compact support onto itself, which commutes with translations can be represented as convolution with a unique Gevrey distribution T of compact support. Finally, we find explicit formula for deconvolvers when the convolvers satisfy weaker conditions than the equivalence conditions using nonperiodic sampling method. [ABSTRACT FROM AUTHOR]

We verify that there exist no gaps between Gevrey differentiable and nowhere Gevrey differentiable in the sense that for given $s>1$, there is a nowhere Gevrey differentiable function on $\mathbb{R}$ of order $s$ that is Gevrey differentiable of order $r$ for any $r>s$, which also gives a strong example that is Gevrey differentiable but nowhere analytic. [ABSTRACT FROM AUTHOR]

We prove the Bochner–Schwartz type theorem for conditionally positive definite Fourier hyperfunctions which generalizes the result of Gelfand-Vilenkin in their treatise Generalized functions, vol. IV for distributions. [ABSTRACT FROM AUTHOR]

We prove the Bochner–Schwartz type theorem for conditionally positive definite Fourier hyperfunctions which generalizes the result of Gelfand-Vilenkin in their treatise Generalized functions, vol. IV for distributions. [ABSTRACT FROM AUTHOR]

Discusses how the Picard's successive approximation of an ordinary differential equation (ODE) converges in the Gevrey sense to the solution of ODE. Occurrence of convergence if the given vector field is a Gevrey mapping; Occurrence of analytic convergence if the vector field is analytic.

We characterize the almost periodic hyperfunctions by showing that the following statements are equivalent for any bounded hyperfunction $T$. (i) $T$ is almost periodic. (ii) $T*\varphi \in C_{ap}$ for every $\varphi \in \mathcal{F}$. (iii) There are two functions $f,g \in C_{ap}$ and an infinite order differential operator $P$ such that $T=P(D^{2})f+g.$ (iv) The Gauss transform $u(x,t)=T*E(x,t)$ of $T$ is almost periodic for every $t>0$. Here $C_{ap}$ is the space of almost periodic continuous functions, $\mathcal{F}$ is the Sato space of test functions for the Fourier hyperfunctions, and $E(x,t)$ is the heat kernel. This generalizes the result of Schwartz on almost periodic distributions and that of Cioranescu on almost periodic (non-quasianalytic) ultradistributions to the case of hyperfunctions. [ABSTRACT FROM AUTHOR]

Every periodic hyperfunction is a bounded hyperfunction and can be represented as an infinite sum of derivatives of bounded continuous periodic functions. Also, Fourier coefficients $c_{\alpha }$ of periodic hyperfunctions are of infra-exponential growth in $\mathbb{R}^{n}$, i.e., $c_{\alpha }< C_{\epsilon }e^{\epsilon |\alpha |}$ for every $\epsilon >0$ and every $\alpha \in \mathbb{Z}^{n}$. This is a natural generalization of the polynomial growth of the Fourier coefficients of distributions. To show these we introduce the space $\mathcal{B}_{L^{p}}$ of hyperfunctions of $L^{p}$ growth which generalizes the space $\mathcal{D}'_{L^{p}}$ of distributions of $L^{p}$ growth and represent generalized functions as the initial values of smooth solutions of the heat equation. [ABSTRACT FROM AUTHOR]

We give an elementary proof of the equivalence of the original definition of Schwartz and our characterization for the Schwartz space $\mathcal{S}$. The new proof is based on the Landau inequality concerning the estimates of derivatives. Applying the same method, as an application, we give a better symmetric characterization of the Beurling--Björck space of test functions for tempered ultradistributions with respect to Fourier transform without conditions on derivatives. [ABSTRACT FROM AUTHOR]

We give symmetric characterizations, with respect to the Fourier transformation, of the Gelfand-Shilov spaces of (generalized) type $S$ and type $W$. These results explain more clearly the invariance of these spaces under the Fourier transformations. [ABSTRACT FROM AUTHOR]