In this paper, we turn to the following Cauchy problem for semilinear models with power nonlinearity and integrable decaying speed of propagation:(1){ϕtt−a(t)2Δϕ+m1a(t)A(t)ϕt+m22(a(t)A(t))2ϕ=|ϕ|p,(t,x)∈(0,∞)×Rn,(ϕ(0,x),ϕt(0,x))=(f(x),g(x)). Here A(t)=∫t∞a(τ)dτ, m1 and m2 are positive constants. The model (1) generalizes the de Sitter model (a(t)=e−t). We are interested to study the interplay of m1 and m2 on the global (in time) well-posedness of small data Sobolev solutions. First of all we propose a classification of models (1). Then our considerations are devoted to two of the models (mass dominant and dissipation dominant in the case of non-effective dissipation). We will see that a large m2 produces a loss of critical exponent pcrit>1.

We study a wide class of space-time fractional equations with structural damping and nonlinear memory: ∂ ^2α_t u(t,x)+μ (- )^σ/2∂ ^α _t u(t,x)+(- )^σ u(t,x)=I^1-α_0^+f(t,x), t≥ 0 , x∈𝐑^n , where the parameters satisfying 1/2 <α≤ 1 , μ >0 and σ > 1 . We take the so-called Caputo-Djrbashian derivative with respect to time t, ∂ _t^κ of order κ with κ =2α , α , the fractional Laplacian in the space variable x, (- )^δ with δ =σ/2,σ , and the Riemann-Liouville integral I^1-α_0^+f(t,x) . We first obtain solution representations for Cauchy type problems expressed in terms of Mittag-Leffler’s functions E_α ,β(z) ( z∈ℂ ) by using a Duhamel type formula with the aid of Laplace (in time) and Fourier (in space) transforms. Taking into account that the Fourier multipliers are radially symmetric functions, the convolution kernels are represented through the Hankel transform. Then, our purpose is to establish decay estimates of the derivatives of its solutions for initial data with L^p and Sobolev regularity. We highlight that the strategy to be employed relies on harmonic analysis tools. We also provide applications employing Strichartz’s estimates to prove existence results for mild solutions in particular cases, in which f(t,x)=| u(t,x) |^k with k >1 or f belongs to a time weighted Banach space.

We consider the nonlinear massless wave equation belonging to some family of the Friedmann–Lemaître–Robertson–Walker (FLRW) spacetime. We prove the global in time small data solutions for supercritical powers in the case of decelerating expansion universe.

We consider the Cauchy problem on $$\mathbf {R_{0}^+} \times \mathbf {R^n}$$ for the semilinear damped wave equation $$\displaystyle u_{tt}(t,x) - a^2(t) \Delta u(t,x) + b(t) u_t(t,x)= |u(t,x)|{ }^p $$ with decreasing in time coefficients, the propagation speed a(t) = (1 + t)−ℓ, ℓ ∈ (0, 1), the scale-invariant dissipation b(t) = β(1 + t)−1, β > 0, and a power nonlinearity of order p > 1. The solution u 0 of the corresponding linear Cauchy problem will be represented in the explicit form using Fourier multipliers operators with multipliers expressed in terms of special functions. Our main goal is to prove a global in time existence result when initial data belongs to the space H m(R n) × H m−1(R n), m ≥ 1. We are focused in finding the critical exponent p c(n, ℓ) such that if 1 < p < p c(n, ℓ) there exist small data for which u blow-up in finite time. We also prove that if p ≥ p c(n, ℓ) the global solution has the same long time behavior as u 0. In order to estimate u we use Duhamel’s principle to represent u and then we apply L 2 − L 2 estimates of u 0.

We consider the generalized Goursat–Darboux problem for a third-order linear PDE with real coefficients. Our purpose is to find necessary conditions for the problem to be well-posed in the Gevrey classes Γs with s > 1. It is proved that there exists some critical index s0 such that if the Goursat–Darboux problem is well-posed in Γs for s > s0, then some conditions should be imposed on the coefficients of the derivatives with respect to one of the variables. In order to prove our results, we first construct an explicit solution of a family of problems with data depending on a parameter η > 0 and then we obtain an asymptotic representation of a solution as η tends to infinity.

We consider the generalized Goursat-Darboux problem for a third order linear PDE with real constant coefficients. Our purpose is to find necessary conditions for the problem to be well-posed in the Gevrey classes. Since this problem can be reduced to the Cauchy problem using permutations of independent variables, we solve it for a ODE with complex coefficients and two unknown initial data. In order to prove our results, we first construct an explicit solution of a family of problems with initial data depending on a parameter η > 0 and then we obtain an asymptotic representation of a solution as η tends to infinity.

This paper deals with the solvability near the characteristic set \({\Sigma = \{0\} \times S^{1}}\) of operators of the form \({L= \partial / \partial t + (x^{n}a(x) + ixb(x))\partial / \partial x, b(0) \neq 0 \,\,{\rm and}\,\, n \geq 2, \,\,{\rm defined \,\, on}\,\, \Omega_{\epsilon} = (-\epsilon,\epsilon) \times S^{1}, \epsilon > 0,}\) where a and b are real-valued smooth functions in \({(-\epsilon,\epsilon). \,\,{\rm For \,\, fixed}\,\, k \geq 1}\), it is shown that given f belonging to a subspace of finite codimension (depending on k) of \({C^{\infty}(\Omega_{\epsilon})}\) there is a solution \({u \in C^{k}}\) of the equation Lu = f in a neighborhood of \({\Sigma}\).

The aim of this paper is to define consumer’s preferences from the differentiable point of view in the sense of Debreu. In this setting we can consider a marginal rate of substitution to represent consumer’s preferences on the space of two goods. By definition, for each point we have a vector that gives the direction orthogonal to an unknown indifference curve. We solve a first order ordinary differential equation in order to determine the indifference map. Our results are representation theorems for the following classes: linear, quasi-linear, separable, homothetic, homothetic and separable. We show that this alternative approach is connected with the formulation concerning the representability of preferences by smooth utility functions. Moreover, we deduce the general expression of utility functions in the context of ordinal utility. Key–Words: smooth preferences, marginal rates of substitution, differential equations, indifference map, ordinal utility.