### A Cauchy problem for ${u}_{t}-\Delta u={u}^{p}\phantom{\rule{4pt}{0ex}}\text{with}\phantom{\rule{4pt}{0ex}}0\<p\<1$. Asymptotic behaviour of solutions

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It is shown that the partial differential operator $P\left(D\right)=\partial \u2074/\partial x\u2074-\partial \xb2/\partial y\xb2+i\partial /\partial z:{\Gamma}^{d}\left(\mathbb{R}\xb3\right)\to {\Gamma}^{d}\left(\mathbb{R}\xb3\right)$ is surjective if 1 ≤ d < 2 or d ≥ 6 and not surjective for 2 ≤ d < 6.

We apply Cartan’s method of equivalence to find a Bäcklund autotransformation for the tangent covering of the universal hierarchy equation. The transformation provides a recursion operator for symmetries of this equation.

We consider a solution u of the homogeneous Dirichlet problem for a class of nonlinear elliptic equations in the form A(u) = g(x,u) + f, where the principal term is a Leray-Lions operator defined on W01,p (Ω). The function g(x,u) satisfies suitable growth assumptions, but no sign hypothesis on it is assumed. We prove that the rearrangement of u can be estimated by the solution of a problem whose data are radially symmetric.

We study a two-particle quantum system given by a test particle interacting in three dimensions with a harmonic oscillator through a zero-range potential. We give a rigorous meaning to the Schrödinger operator associated with the system by applying the theory of quadratic forms and defining suitable families of self-adjoint operators. Finally we fully characterize the spectral properties of such operators.