Generalized summability results are obtained regarding formal solutions of certain families of linear moment integro-differential equations with time variable coefficients. The main result leans on the knowledge of the behavior of the moment derivatives of the elements involved in the problem.
A refinement of the main result is also provided giving rise to more accurate results which remain valid in wide families of problems of high interest in practice, such as fractional integro-differential equations.
We study the Cauchy problem for a general homogeneous linear partial differential equation in two complex variables with constant coefficients and with divergent initial data. We state necessary and sufficient conditions for the summability of formal power series solutions in terms of properties of divergent Cauchy data. We consider both the summability in one variable t (with coefficients belonging to some Banach space of Gevrey series with respect to the second variable z) and the summability in two variables (t,z). The results are presented in the general framework of moment-PDEs.
The concept of moment differentiation is extended to the class of moment summable functions, giving rise to moment differential properties. The main result leans on accurate upper estimates for the integral representation of the moment derivatives of functions under exponential-like growth at infinity, and appropriate deformation of the integration paths. The theory is applied to obtain summability results of certain family of generalized linear moment partial differential equations with variable coefficients.
We deal with a class of equations driven by nonlocal, possibly degenerate, integro-differential operators of differentiability order $s\in (0,1)$ and summability growth $p>1$, whose model is the fractional $p$-Laplacian with measurable coefficients. We state and prove several results for the corresponding weak supersolutions, as comparison principles, a priori bounds, lower semicontinuity, and many others. We then discuss the good definition of $(s,p)$-superharmonic functions, by also proving some related properties. We finally introduce the nonlocal counterpart of the celebrated Perron method…
The paper shows the summability of formal solutions of some linear q-difference-differential equations by using q-Laplace and q-Borel summation method.
The paper discusses the summability of formal solutions of some linear q-difference-differential equations, and improves the previous result in [Tahara-Yamazawa, Opsucula Math. 35 (2015), 713-738].
In this paper, we study a class of first order nonlinear degenerated partial differential equations with singularity at $(t,x)=(0,0)\in \CC^2$. By means of exponential type Nagumo norm approach, Gevrey asymptotic analysis extends to case of holomorphic parameters by a natural way. A sharp condition is then established to deduce the $k$-summability for the formal solutions. Furthermore, analytical solutions in conical domains are found for each type of these nonlinear singular PDEs.
We study the Cauchy problem for a general inhomogeneous linear moment partial differential equation of two complex variables with constant coefficients, where the inhomogeneity is given by the formal power series. We state sufficient conditions for the convergence, analytic continuation and summability of formal power series solutions in terms of properties of the inhomogeneity. We consider both the summability in one variable t (with coefficients belonging to some Banach space of Gevrey series with respect to the second variable z) and the summability in two variables (t,z).
We investigate Gevrey order and 1-summability properties of the formal solution of a general heat equation in two variables. In particular, we give necessary and sufficient conditions for the 1-summability of the solution in a given direction. When restricted to the case of constants coefficients, these conditions coincide with those given by D.A. Lutz, M. Miyake, R. Schaefke in a 1999 article, and we thus provide a new proof of their result.
Through Borel summation methods, we analyze two different variations of the Navier-Stokes equation --the Boussinesq equation and the magnetic Benard equation. This method has previously been applied to the Navier-Stokes equation. We prove that an equivalent system of integral equations in each case has a unique solution, which is exponentially bounded for p in R^{+}, p being the Laplace dual variable of 1/t. This implies the local existence of a classical solution in a complex t-region that includes a real positive time (t)-axis segment. In this formalism, global existence of PDE solutions bec…
The periodic homogenization problem of integro-differential equations of the alpha stable L{é}vy operators is studied in this paper. Thanking to the symmetry of the L{é}vy density, we can use the method of the formal asymptotic expansion, to connect the problem to the ergodic cell problem. A rigorous proof is given by the perturbed test function's method.
Through Borel summation methods, we analyze the Boussinesq equations for coupled fluid velocity and temperature fields. We prove that an equivalent system of integral equations in the Borel variable p dual to 1/t has a unique solution in a class of exponentially bounded functions, implying the existence of a classical solution to the Boussinesq equations in a complex t-region that includes a real positive time axis segment. For analytic initial data and forcing, it is shown that the solution is Borel summable, implying that that formal series in powers of t is Gevrey-1 asymptotic, and within t…
We consider periodic homogenization problems for the L{é}vy operators with asymmetric L{é}vy densities. The formal asymptotic expansion used for the $\a$-stable (symmetric) L{é}vy operators ($\a\in (0,2)$) is not applicable directly to such asymmetric cases. We rescale the asymmetric densities, extract the most singular part of the measures, which average out the microscopic dependences in the homogenization procedures. We give two conditions (A) and (B), which characterize such a class of asymmetric densities, that the above "rescaled" homogenization is available.
We describe partial differential operators for which we can construct generalised integral means satisfying Pizzetti-type formulas. Using these formulas we give a new characterisation of summability of formal power series solutions to some multidimensional partial differential equations in terms of holomorphic properties of generalised integral means of the Cauchy data.
We consider a system of $n$-th order nonlinear quasilinear partial differential equations of the form $${\bf u}_t + \mathcal{P}(\partial_{\bf x}^{\bf j}){\bf u}+{\bf g} \left( {\bf x}, t, \{\partial_{\bf x}^{\bf j} {\bf u}\}) =0; {\bf {u}}({\bf x}, 0) = {\bf {u}}_I({\bf x})$$ with $\mathbf{u}\in\CC^{r}$, for $ t\in (0,T)$ and large $|{\bf x}|$ in a poly-sector $S$ in $\mathbb{C}^d$ ($\partial_{\bf x}^{\bf j} \equiv \partial_{x_1}^{j_1} \partial_{x_2}^{j_2} ...\partial_{x_d}^{j_d}$ and $j_1+...+j_d\le n$). The principal part of the constant coefficient $n$-th order differential operator $\mathc…
We consider integro-differential models describing the evolution of a population structured by a quantitative trait. Individuals interact competitively, creating a strong selection pressure on the population. On the other hand, mutations are assumed to be small. Following the formalism of Diekmann, Jabin, Mischler, and Perthame, this creates concentration phenomena, typically consisting in a sum of Dirac masses slowly evolving in time. We propose a modification to those classical models that takes the effect of small populations into accounts and corrects some abnormal behaviours.
We consider a broad class of systems of nonlinear integro-differential equations posed on the real line that arise as Euler-Lagrange equations to energies involving nonlinear nonlocal interactions. Although these equations are not readily cast as dynamical systems, we develop a calculus that yields a natural Hamiltonian formalism. In particular, we formulate Noether's theorem in this context, identify a degenerate symplectic structure, and derive Hamiltonian differential equations on finite-dimensional center manifolds when those exist. Our formalism yields new natural conserved quantities. Fo…
We consider a integro-differential nonlinear model that describes the evolution of a population structured by a quantitative trait. The interactions between traits occur from competition for resources whose concentrations depend on the current state of the population. Following the formalism of\cite{DJMP}, we study a concentration phenomenon arising in the limit of strong selection and small mutations. We prove that the population density converges to a sum of Dirac masses characterized by the solution $ϕ$ of a Hamilton-Jacobi equation which depends on resource concentrations that we fully cha…
We present a spatial hybrid discrete-continuum modelling framework for the interaction dynamics between tumour cells and cytotoxic T cells, which play a pivotal role in the immune response against tumours. In this framework, tumour cells and T cells are modelled as individual agents while chemokines that drive the chemotactic movement of T cells towards the tumour are modelled as a continuum. We formally derive the continuum counterpart of this model, which is given by a coupled system that comprises an integro-differential equation for the density of tumour cells, a partial differential equat…
