On the structure and stability of the set of solutions of a nonlocal problem modeling Ohmic heating (Q1272716)

The author studies the existence and stability of the positive steady states of the nonlocal problem \[ \partial_tu+{\mathcal L}u= \lambda{f(u)\over [a+ b\int_\Omega f(u(x)) dx]^2}\quad\text{in }\Omega\times (0,\infty),\tag{1} \] \[ {\mathcal B}u= 0\quad\text{on }\partial\Omega\times (0,\infty),\quad u(\cdot,0)= u_0\geq 0\quad\text{in }\overline\Omega,\tag{2} \] where \({\mathcal L}\) is a second-order uniformly elliptic operator, \(a\geq 0\), \(b\geq 0\); \(\lambda\geq 0\) is a parameter, \(f: [0,\infty)\mapsto (0,\infty)\) is of class \(C^1\) and \(f'(u)<0\) for all \(u\in [0,\infty)\). Sufficient conditions are obtained so that the problem (1)--(2) admits a positive steady state for each value of \(\lambda\).