We consider the equation $$\label {1} - (r(x)y'(x))'+q(x)y(x)=f(x),\quad x\in \mathbb R \eqno {(*)} $$ where $f\in L_p(\mathbb R)$, $p\in (1,\infty )$ and \begin {gather} r>0,\quad q\ge 0,\quad \frac {1}{r}\in L_1^{\rm loc}(\mathbb R),\quad q\in L_1^{\rm loc}(\mathbb R), \nonumber \\ \lim _{|d|\to \infty }\int _{x-d}^x \frac {{\rm d} t}{r(t)}\cdot \int _{x-d}^x q(t) {\rm d} t=\infty . \nonumber \end {gather} In an earlier paper, we obtained a criterion for correct solvability of ($*$) in $L_p(\mathbb R),$ $p\in (1,\infty ).$ In this criterion, we use values of some auxiliary implicit functions in the coefficients $r$ and $q$ of equation ($*$). Unfortunately, it is usually impossible to compute values of these functions. In the present paper we obtain sharp by order, two-sided estimates (an estimate of a function $f(x)$ for $x\in (a,b)$ through a function $g(x)$ is sharp by order if $c^{-1}|g(x)|\le |f(x)|\le c|g(x)|,$ $x\in (a,b),$ $c=\rm const$) of auxiliary functions, which guarantee efficient study of the problem of correct solvability of ($*$) in $L_p(\mathbb R),$ $p\in (1,\infty ).$.
The paper is devoted to the analysis of an abstract evolution inclusion with a non-invertible operator, motivated by problems arising in nonlocal phase separation modeling. Existence, uniqueness, and long-time behaviour of the solution to the related Cauchy problem are discussed in detail.