## A transport model with adsorption hysteresis.(English)Zbl 1004.35033

The paper deals with an initial-boundary value problem for the transport equation $$(u+v)_t + u_x = 0$$, $$v = {\mathcal M}u$$, $$u(0,t) = \varphi (t)$$, $$u(x,0) = v(x,0) = 0$$, $$t \in [0,T]$$, $$x \in [0,L]$$, where $$\mathcal M$$ is a hysteresis operator, more specifically the play operator or its combinations (the so-called Prandtl-Ishlinskij operator). Considering alternatively $$x$$ and $$t$$ as the ‘time’ variable and rewriting the problem as an abstract evolution equation of the form $$U' + AU \ni f$$ in $$L^2 \times L^2$$ with an $$m$$-accretive operator $$A$$, the authors obtain two different existence, uniqueness and regularity results. In particular, as it is often the case in hyperbolic PDEs with convex hysteresis, shocks do not occur.

### MSC:

 35F20 Nonlinear first-order PDEs 47H06 Nonlinear accretive operators, dissipative operators, etc. 47J35 Nonlinear evolution equations