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$(u,L\varphi)_0-(Vu,\varphi)_0+(g(\cdot,u,\nabla u),\varphi)_0=\mu(\varphi),\quad\forall\varphi\in C^2_c(\Omega).$

The potential $V \le \lambda < \lambda_1$ is assumed to be in the
weighted Lorentz space $L^{N,1}(\Omega,\delta)$, where
$\delta(x)= dist(x,\partial\Omega),\ \mu\in
M^1(\Omega,\delta)$, the set of weighted Radon measures
containing $L^1(\Omega,\delta)$, $L$ is an elliptic linear self
adjoint second order operator, and $\lambda_1$ is the first
eigenvalue of $L$ with zero Dirichlet boundary conditions.

If $\mu\in L^1(\Omega,\delta)$ we only assume that for the potential $V$ is in
L^{1}_{loc}$(\Omega)$, $V \le \lambda<\lambda_1$. If $\mu\in M^1(\Omega,\delta^\alpha),\
\alpha\in$[$0,1[$[, then we prove that the very weak solution $|\nabla u|$ is in the
Lorentz space $L^{\frac N{N-1+\alpha},\infty}(\Omega)$. We apply those results
to the existence of the so called large solutions with a right hand side data in
$L^1(\Omega,\delta)$. Finally, we prove some rearrangement comparison results.

In this paper we prove the existence and uniqueness of very weak solutions to linear diffusion equations involving a singular absorption potential and/or an unbounded convective flow on a bounded open set of $\text{IR}^N$. In most of the paper we consider homogeneous Dirichlet boundary conditions but we prove that when the potential function grows faster than the distance to the boundary to the power -2 then no boundary condition is required to get the uniqueness of very weak solutions. This result is new in the literature and must be distinguished from other previous results in which such uniqueness of solutions without any boundary condition was proved for degenerate diffusion operators (which is not our case). Our approach, based on the treatment on some distance to the boundary weighted spaces, uses a suitable regularity of the solution of the associated dual problem which is here established. We also consider the delicate question of the differentiability of the very weak solution and prove that some suitable additional hypothesis on the data is required since otherwise the gradient of the solution may not be integrable on the domain.

$\partial_t u - \Delta_N u=\mu$ in $\mathcal D'(Q) $

$u=0$ on $]0,T[\times\partial \Omega$

$u(0)=u_0$ in $ \Omega,$

where $Q$ is the cylinder $Q=(0,T)\times\Omega$, $T>0$, $\Omega\subset \mathbb R^n$, $N\ge 2$, is an open bounded set having $C^2$ boundary, $\mu\in L^1(0,T;M(\Omega))$ and $u_0$ belongs to $M(\Omega)$, the space of the Radon measures in $\Omega$, or to $L^1(\Omega)$. The results are obtained in the framework of the so-called grand Sobolev spaces, and represent an extension of earlier results on standard Sobolev spaces.

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