### A link between ${C}^{\infty}$ and analytic solvability for P.D.E. with constant coefficients

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If $P$ is a polynomial in ${\mathbf{R}}^{n}$ such that $1/P$ integrable, then the inverse Fourier transform of $1/P$ is a fundamental solution ${E}_{P}$ to the differential operator $P\left(D\right)$. The purpose of the article is to study the dependence of this fundamental solution on the polynomial $P$. For $n=1$ it is shown that ${E}_{P}$ can be analytically continued to a Riemann space over the set of all polynomials of the same degree as $P$. The singularities of this extension are studied.

Let A(Ω) denote the real analytic functions defined on an open set Ω ⊂ ℝⁿ. We show that a partial differential operator P(D) with constant coefficients is surjective on A(Ω) if and only if for any relatively compact open ω ⊂ Ω, P(D) admits (shifted) hyperfunction elementary solutions on Ω which are real analytic on ω (and if the equation P(D)f = g, g ∈ A(Ω), may be solved on ω). The latter condition is redundant if the elementary solutions are defined on conv(Ω). This extends and improves previous...

Let $K\subset Q$ be compact, convex sets in ${\mathbb{R}}^{n}$ with $\stackrel{\circ}{\phantom{\rule{0.0pt}{0ex}}K}\ne \varnothing $ and let $P\left(D\right)$ be a linear, constant coefficient PDO. It is characterized in various ways when each zero solution of $P\left(D\right)$ in the space $\mathcal{E}\left(K\right)$ of all ${C}^{\infty}$-functions on $K$ extends to a zero solution in $\mathcal{E}\left(Q\right)$ resp. in $\mathcal{E}\left({\mathbb{R}}^{n}\right)$. The most relevant characterizations are in terms of Phragmén-Lindelöf conditions on the zero variety of $P$ in ${\u2102}^{n}$ and in terms of fundamental solutions for $P\left(D\right)$ with lacunas.

We prove the ${L}^{p}$-${L}^{q}$-time decay estimates for the solution of the Cauchy problem for the hyperbolic system of partial differential equations of linear thermoelasticity. In our proof based on the matrix of fundamental solutions to the system we use Strauss-Klainerman’s approach [12], [5] to the ${L}^{p}$-${L}^{q}$-time decay estimates.

This note is devoted to the study of the long time behaviour of solutions to the heat and the porous medium equations in the presence of an external source term, using entropy methods and self-similar variables. Intermediate asymptotics and convergence results are shown using interpolation inequalities, Gagliardo-Nirenberg-Sobolev inequalities and Csiszár-Kullback type estimates.

Let P(D) be a partial differential operator with constant coefficients which is surjective on the space A(Ω) of real analytic functions on an open set $\Omega \subset {\mathbb{R}}^{n}$. Then P(D) admits shifted (generalized) elementary solutions which are real analytic on an arbitrary relatively compact open set ω ⊂ ⊂ Ω. This implies that any localization ${P}_{m,\Theta}$ of the principal part ${P}_{m}$ is hyperbolic w.r.t. any normal vector N of ∂Ω which is noncharacteristic for ${P}_{m,\Theta}$. Under additional assumptions ${P}_{m}$ must be locally hyperbolic.