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The purpose of this paper is to give a necessary and sufficient condition for the existence and non-existence of global solutions of the following semilinear parabolic equations \[ u_{t}=\Delta u+\psi(t)f(u),\,\,\mbox{ in }\Omega\times (0,t^{*}), \] under the Dirichlet boundary condition on a bounded domain. In fact, this has remained as an open problem for a few decades, even for the case $f(u)=u^{p}$. As a matter of fact, we prove: \[ \begin{aligned} &\mbox{there is no global solution for any initial data if and only if } &\mbox{the function } f \mbox{ satisfies} &\hspace{20mm}\int_{0}^{\infty}\psi(t)\frac{f\left(\lVert S(t)u_{0}\rVert_{\infty}\right)}{\lVert S(t)u_{0}\rVert_{\infty}}dt=\infty &\mbox{for every }\,\epsilon>0\,\mbox{ and nonnegative nontrivial initial data }\,u_{0}\in C_{0}(\Omega). \end{aligned} \] Here, $(S(t))_{t\geq 0}$ is the heat semigroup with the Dirichlet boundary condition., Comment: 12 pages

In this paper, we are interested in long time behaviors of solutions to the discrete Laplacian parabolic equations u t = Δ ω u + ψ f ( u ) with nonnegative and non-trivial initial data. In particular, we assume that the function f is convex only on a short interval and f ( α s ) ≈ f ( α ) f ( s ) for 0 α 1 , s > 0 , and we present a critical set depending on the function ψ in the sense that if f is in the critical set, then solutions blow up in finite time for any initial data, and if not, then solutions are global or blow up according to the size of initial data.

In this paper, we consider the following discrete p-Laplacian nonlinear parabolic equations with q-nonlocal reaction { u t ( x , t ) = Δ p , ω u ( x , t ) + λ ∑ y ∈ S | u ( y , t ) | q − 1 u ( y , t ) , ( x , t ) ∈ S × ( 0 , ∞ ) , u ( x , t ) = 0 , ( x , t ) ∈ ∂ S × ( 0 , ∞ ) , u ( x , 0 ) = u 0 ≥ 0 , x ∈ S ‾ . Here, S is a network with boundary ∂S. The goal of this paper is to characterize completely the parameters p > 1 and q > 0 to see when the solution blows up, vanishes, or exists globally. Indeed, the blow-up rates for the blow-up solutions are derived.

In this paper, we consider an initial-boundary value problem of the p-Laplacian parabolic equations \begin{equation} \begin{cases} u_{t}\left(x,t\right)=\mbox{div}(|\nabla u\left(x,t\right)|^{p-2}\nabla u(x,t))+f(u(x,t)), & \left(x,t\right)\in \Omega\times\left(0,+\infty\right), \newline u\left(x,t\right)=0, & \left(x,t\right)\in\partial \Omega\times\left[0,+\infty\right), \newline u\left(x,0\right)=u_{0}\geq0, & x\in\overline{\Omega}, \end{cases} \end{equation} where $p\geq2$ and $\Omega$ is a bounded domain of $\mathbb{R}^{N}$ $(N\geq1)$ with smooth boundary $\partial\Omega$. The main contribution of this work is to introduce a new condition \[ \mbox{$(C_{p})$$\hspace{1cm} \alpha \int_{0}^{u}f(s)ds \leq uf(u)+\beta u^{p}+\gamma,\,\,u>0$} \] for some $\alpha, \beta, \gamma>0$ with $0, Comment: 15 pages. arXiv admin note: text overlap with arXiv:1706.03494

In this paper, we discuss and answer the following dichotomy problems: Let S be a network and Δ p , ω be a discrete p -Laplace operator with 1 p ∞ . (i) If u , v are functions satisfying − Δ p , ω u x ≤ − Δ p , ω v x , x ∈ S , u z ≤ v z , z ∈ ∂ S , then either u ≡ v on S ¯ or u v in S . (ii) If u , v are functions satisfying u t x , t − Δ p , ω u x , t ≤ v t x , t − Δ p , ω v x , t , x , t ∈ S × 0 , T , u x , 0 ≤ v x , 0 , x ∈ S , u z , t ≤ v z , t , z , t ∈ ∂ S × 0 , T then either u ≡ v on S ¯ or u v in S × ( 0 , T ) . We believe that this work is not only interesting in itself, but also gives a clue to solve the problems defined on the continuous domain.

This work is concerned with the extinction and positivity properties of solutions to the discrete p-Laplacian parabolic equation u t = Δ p , ω u − f ( u ) , p > 1 with a nonlinear absorption on a weighted graph. We provide a complete characterization for f to see when we have an extinctive solution or a positive solution. In addition, the extinction time is estimated for the extinctive solutions, and the decay rate is given to the positive solutions. Finally, we give several numerical examples which illustrate the main results.

The purpose of this paper is to investigate a condition \begin{equation*} (C_{p}) \hspace{1cm} \alpha \int_{0}^{u}f(s)ds \leq uf(u)+\beta u^{p}+\gamma,\,\,u>0 \end{equation*} for some $\alpha>2$, $\gamma>0$, and $0\leq\beta\leq\frac{\left(\alpha-p\right)\lambda_{p,0}}{p}$, where $p>1$ and $\lambda_{p,0}$ is the first eigenvalue of the discrete $p$-Laplacian $\Delta_{p,\omega}$. Using the above condition, we obtain blow-up solutions to discrete $p$-Laplacian parabolic equations \begin{equation*} \begin{cases} u_{t}\left(x,t\right)=\Delta_{p,\omega}u\left(x,t\right)+f(u(x,t)), & \left(x,t\right)\in S\times\left(0,+\infty\right), \mu(z)\frac{\partial u}{\partial_{p} n}(x,t)+\sigma(z)|u(x,t)|^{p-2}u(x,t)=0, & \left(x,t\right)\in\partial S\times\left[0,+\infty\right), u\left(x,0\right)=u_{0}\geq0(nontrivial), & x\in S, \end{cases} \end{equation*} on a discrete network $S$, where $\frac{\partial u}{\partial_{p}n}$ denotes the discrete $p$-normal derivative. Here, $\mu$ and $\sigma$ are nonnegative functions on the boundary $\partial S$ of $S$, with $\mu(z)+\sigma(z)>0$, $z\in \partial S$. In fact, it will be seen that the condition $(C_{p})$, the generalized version of the condition $(C)$, improves the conditions known so far., Comment: 22 pages. arXiv admin note: substantial text overlap with arXiv:1706.03494

This paper is concerned with long time behaviors of solutions to the initial-boundary value problem of the evolution p -Laplacian equations with nonlinear absorption. A long time behavior of solutions to these equations has been studied so far for particular types of nonlinear absorption term. The purpose of this paper is to give a complete characterization of the nonlinear absorption term, according to the parameter p in p -Laplacian operator and the growth of the nonlinear absorption term near the origin, in order to determine whether the solution to the equations is extinctive or positive. In addition, we also give upper bounds for extinction times of solutions.

In this paper, we discuss the extinction and positivity of solutions to evolution p -Laplacian equations u t - Δ p , ω u + µ | u | q - 1 u = 0 with absorption on finite networks with p > 1 , q > 0 , and µ ? 0 . We present necessary and sufficient conditions for parameters p , q , and µ in order that solutions to the equation are extinctive or positive, respectively. In addition, when a solution is positive, we provide its decay rate and when a solution becomes extinctive in finite time, we give estimate for the extinction time.

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.

The purpose of this paper is to introduce a new condition \[ \hbox{(C)$\hspace{1cm} \alpha \int_{0}^{u}f(s)ds \leq uf(u)+\beta u^{2}+\gamma,\,\,u>0$} \] for some $\alpha, \beta, \gamma>0$ with $0, Comment: 19 pages

LetSbe a commutative semigroup if not otherwise specified andf:S→ℝ. In this paper we consider the stability of exponential functional equations|f(x+σ(y))-g(x)f(y)|≤ϕ(x)or ϕ(y),|f(x+σ(y))-f(x)g(y)|≤ϕ(x)orϕ(y)for allx,y∈Sand whereσ:S→Sis an involution. As main results we investigate bounded and unbounded functions satisfying the above inequalities. As consequences of our results we obtain the Ulam-Hyers stability of functional equations (Chung and Chang (in press); Chávez and Sahoo (2011); Houston and Sahoo (2008); Jung and Bae (2003)) and a generalized result of Albert and Baker (1982).

We address some forward and inverse problems involving indefinite eigenvalues for discretep-Laplacian operators with potential terms. These indefinite eigenvalues are the discrete analogues ofp-Laplacians on Riemannian manifolds with potential terms. We first define and discuss some fundamental properties of the indefinite eigenvalue problems for discretep-Laplacian operators with potential terms with respect to some given weight functions. We then discuss resonance problems, anti-minimum principles, and inverse conductivity problems for the discretep-Laplacian operators with potential terms involving the smallest indefinite eigenvalues.

In this paper, we consider a discrete version of the following p-Laplacian evolution equation u t = Δ p u , with p > 1 , on a network. Using the discrete p-Laplacian operator in graphs which means a nonlinear diffusion phenomenon on networks, we first introduce the p-Laplacian evolution equation on networks. In fact, spatial derivative in p-Laplacian evolution equation, comparing the continuous case, is replaced with the discrete p-Laplacian operator. Thus, the resulting system is semi-discrete, discrete in space and continuous in time. The main concern is to investigate the large time behaviors of nontrivial solutions of this equation whose initial data are nonnegative and the boundary data vanish. In order to do so, we use the analytic approaches such as vector calculus on networks, maximum principle and comparison principle instead of numerical ones. After deriving the basic properties of this equation, we finally prove that the solution becomes extinct for 1 p 2 and remains strictly positive for p ⩾ 2 .

In this paper, we deal with the discrete p-Laplacian opera- tors with a potential term having the smallest nonnegative eigenvalue. Such operators are classied as its smallest eigenvalue is positive or zero. We discuss differences between them such as an existence of solutions of p-Laplacian equations on networks and properties of the energy func- tional. Also, we give some examples of Poisson equations which suggest a difference between linear types and nonlinear types. Finally, we study characteristics of the set of a potential those involving operator has the smallest positive eigenvalue.

In this paper, we discuss the existence and the uniqueness of discrete boundary value problems involving the p-Laplacian with potential terms. To this end, we deal with the strong comparison principle and the Dirichlet principle for the p-Laplacian with potential terms on weighted graphs with boundary. Moreover, we provide a lower bound of the potential terms which guarantees the existence of solutions of the boundary value problem for the operator. Finally, we also discuss inverse conductivity and potential problems for the operator.

The purpose of this work is to identify the internal structure of the electrical networks with data obtained from only a part of network or the boundary of network. To be precise, it is discussed whether we can identify resistors or electrical conductivities of each link inside networks by the measurement of voltage on the boundary which is induced by a prescribed current on the boundary. As a result, it is shown that the structure of the resistor network can be determined uniquely by only one pair of the data (current, voltage) on the boundary, if the resistors satisfy an appropriate condition. Besides, several useful results about the energy functionals, which means the electrical power, are included.

In this paper, we characterize Sobolev spaces Hs(Rn), s ∈ R by the initial value of solutions of heat equation with a growth condition. By using an idea in its proof, we also discuss a stability problem for Cauchy functional equation in the Sobolev spaces.

In this paper, we consider the general solution of quadratic functional equation f(ax+y)+f(ax−y)=f(x+y)+f(x−y)+2(a2−1)f(x) for any integer a with a≠−1,0,1. Moreover we reformulate and prove the Hyers–Ulam–Rassias stability theorem of the above equation in the spaces of tempered distributions and Fourier hyperfunctions. The generalized Hyers–Ulam stability originated from the Th.M. Rassias's stability theorem that appeared in his paper [Th.M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978) 297–300].

In this paper, we discuss discrete versions of the heat equations and the wave equations, which are called the ω-diffusion equations and the ω-elastic equations on graphs. After deriving some basic properties, we solve the ω-diffusion equations under (i) the condition that there is no boundary, (ii) the initial condition and (iii) the Dirichlet boundary condition. We also give some additional interesting properties on the ω-diffusion equations, such as the minimum and maximum principles, Huygens property and uniqueness via energy methods. Analogues of the ω-elastic equations on graphs are also discussed.

C. Berenstein and the first author introduced an elliptic operator Δω and an ω-harmonic function on graphs, with its physical interpretation as a diffusion equation on graphs, which models electric networks. They also proved the solvability of the problems such as the Dirichlet and Neumann boundary value problems for the Laplace equation and the Poisson equation on networks. In this paper, we consider the Poisson equation with nonlinear source term on networks and show the existence and the uniqueness of a solution with suitable source term.

We prove the Ulam problem for the cosine addition formula in the spaces of Schwartz distributions and Sato hyperfunctions with respect to bounded distributions and bounded hyperfunctions.

We reformulate and solve the stability problem of a Jensen type functional equation 3 f ( x + y + z 3 ) + f ( x ) + f ( y ) + f ( z ) − 2 f ( x + y 2 ) − 2 f ( y + z 2 ) − 2 f ( z + x 2 ) = 0 , in the spaces of some generalized functions such as tempered distributions and Fourier hyperfunctions.

We use Cauchy's functional equation to construct a new non-measurable set which is a (vector) subspace of and is of a codimensiion 1, considering , the set of real numbers, as a vector space over a field of rational numbers. Moreover, we show that can be partitioned into a countable family of disjoint non-measurable subsets.

We consider a generalized Pompeiu equation in the space of Schwartz distributions and as an application we find the locally integrable solutions of the equation.

We reformulate a quadratic functional equation of the form f (x + y + z) + f (x− y + z) + f (x + y − z) + f (−x + y + z) = 4f (x) + 4f (y) + 4f (z) and an inequality |f (x + y + z) + f (x − y + z) + f (x + y − z) + f (−x + y + z) − 4f (x) − 4f (y) − 4f (z)| e in the space of distributions. In view of this fact, we use a mollifier and Gauss transform to show that every distributional solution of the inequality is a tempered distribution and finally the stability problem of the equation in the sense of distributions. Mathematics subject classification (2000): 39B82, 46F12.

In this paper, we discuss the inverse problem of identifying the connectivity and the conductivity of the links between adjacent pair of nodes in a network, in terms of an input-output map. To do t...

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 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.

We present the sampling theorem with sampling functions of general form for entire functions satisfying one of the growth conditions 1+|y| f(z) ⩽A 1+|z| N 1 exp τ|x|+σ|y| , 1+|x| f(z) ⩽A 1+|z| N 1 exp τ|x|+σ|y| for some A>0, τ,σ⩾0, N 1 ∈ N ∪{0} and any z=x+iy∈ C . It will be shown that many well-known sampling theorems included in SIAM J. Math. Anal. 19 (1988) 1198–1203 and Inform. Control 8 (1965) 143–158 can be interpreted as special cases of this sampling theorem. As examples, we provide sampling representations for entire functions which are bounded, of polynomial growth, or of exponential growth on R . We also provide sampling representations involving derivatives of entire functions and nonuniform sampling representations. Taking the set of sampling points in which a finite number of points are arbitrarily distributed, we obtain a sampling representation.

It is shown that for a positive definite continuous function f(x) on Rn the followings are equivalent: (i) f(x) is quasaianalytic in some neighborhood of the origin. (ii) f(x) can be expressed as an integral f(x) = ∫Rn ei x ξ dµ(ξ) for some positive Radon measure µ on Rn such that ∫ exp M(L|ξ|)dµ(ξ) is finite for some L > 0 where the function M(t) is a weight function corresponding to the quasaianalyticity. (iii) f(x) is quasaianalytic everywhere in Rn. Moreover, an analogue for the analyticity is also given as a corollary.

We discuss the conditions under which blow-up occurs for the solutions of discretep-Laplacian parabolic equations on networksSwith boundary∂Sas follows:ut(x,t)=Δp,ωu(x,t)+λ|u(x,t)|q-1u(x,t),(x,t)∈S×(0,+∞);u(x,t)=0,(x,t)∈∂S×(0,+∞);u(x,0)=u0≥0,x∈S¯, wherep>1,q>0,λ>0, and the initial datau0is nontrivial onS. The main theorem states that the solutionuto the above equation satisfies the following: (i) if01, then the solution blows up in a finite time, providedu¯0>ω0/λ1/q-p+1, whereω0:=maxx∈S⁡∑y∈S¯‍ω(x,y)andu¯0:=maxx∈S u0(x); (ii) if0

The goal of this paper is to investigate the blow-up and the global existence of the solutions to the discretep-Laplacian parabolic equationutx,t=Δp,wux,t+λux,tp-2ux,t,x,t∈S×0,∞,ux,t=0,x,t∈∂S×0,∞,ux,0=u0, depending on the parametersp>1andλ>0. Besides, we provide several types of the comparison principles to this equation, which play a key role in the proof of the main theorems. In addition, we finally give some numerical examples which exploit the main results.

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 ∗ φ ∈ Cap for every φ ∈ F . (iii) There are two functions f, g ∈ Cap and an infinite order differential operator P such that T = P (D2)f + g. (iv) The Gauss transform u(x, t) = T ∗ E(x, t) of T is almost periodic for every t > 0. Here Cap is the space of almost periodic continuous functions, 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.

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 α c_{\alpha } of periodic hyperfunctions are of infra-exponential growth in R n \mathbb {R}^{n} , i.e., c α > C ϵ e ϵ | α | c_{\alpha }> C_{\epsilon }e^{\epsilon |\alpha |} for every ϵ > 0 \epsilon >0 and every α ∈ Z n \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 B L p \mathcal {B}_{L^{p}} of hyperfunctions of L p L^{p} growth which generalizes the space D L p ′ \mathcal {D}’_{L^{p}} of distributions of L p L^{p} growth and represent generalized functions as the initial values of smooth solutions of the heat equation.

We relax the growth condition in time for uniqueness of solutions of the Cauchy problem for the heat equation as follows: Let u(x, t) be a continuous function on ℝn × [0, T] satisfying the heat equation in ℝn × (0, t) and the following:(i) There exist constants a > 0, 0 < α < 1, and C > 0 such that(ii) u(x, 0) = 0 for x ∈ ℝn.Then u(x, t)≡ 0 on ℝn × [0, T]We also prove that the condition 0 < α < 1 is optimal.

We introduce an integral transformation T defined by -W f(y)dy, in order to do harmonic analysis on the space of C°° solutions of heat equation. First, the Paley-Wiener type theorem for the transformation T will be given for the C°° functions and distributions with compact support. Secondly, as an application of the transformation the solutions of heat equation given on the torus J" will be characterized. Finally, we represent solutions of heat equation as an infinite series of Hermite temperatures, which are to be defined as the images of Hermite polynomials under the transformation T. §