Diffusion in a fissured medium with absorption or partial saturation effects leads to the initial-boundary value problem \[ (P)\quad u'+\epsilon^{-1}(\alpha (u)-v)=f_ 1,\quad -div(k(u)\nabla v)+\epsilon^{-1}(v-\alpha (u))=f_ 2\quad in\quad Q, \] u(x,0)\(=u_ 0(x)\), \(v_{\partial G}=0\), where \(S=[0,T]\), G is a smooth bounded domain in \(R^ N\) and \(Q=(0,T)\times G.\) Let us consider the following hypotheses: (H1) \(k\in C(R)\) and there are \(k_ 0\), \(k_ 1\) such that \(0\max \{1,N/2],\quad s\in (0,1]\) sufficiently small; \((d)\quad N=p=1,\quad s\in (0,1].\) Then (AP) has a solution \(u\in C^ 1(S,W_ 0^{s,p}(G)).\) If \(f\in L^ r(S,H^ 1_ 0(G))\) and \(u_ 0\in H^ 1_ 0(G)\), then the variational formulation of (AP) has a solution \(u\in W^{1,r}(S,H^ 1_ 0(G));\) if \(f\in C(S,H^ 1_ 0(G)),\) then \(u\in C^ 1(S,H^ 1_ 0(G)).\) Assume (H1), (H2), (H4), (H5), (H6), \(N=2\), \(p\in [2,\infty].\) If \(f\in L^ r(S,W^{2,p}(G)\cap W_ 0^{1,2}(G)),\quad u_ 0\in W^{2,p}(G)\cap W_ 0^{1,p}(G),\) then the solution u of (AP) is in \(W^{1,r}(S,W^{2,p}(G)\cap W_ 0^{1,p}(G));\) if \(f\in C(S,W^{2,p}(G)\cap W_ 0^{1,p}(G)),\) then \(u\in C^ 1(S,W^{2,p}(G)\cap W_ 0^{1,p}(G)).\) Assume (H1), (H4), (H6). So \(p>2\) if \(N=2\) and \(p=N\) otherwise. If \(u_ 0\in H^ 1_ 0(G)\), \(f\in L^ 1(S,W_ 0^{1,p}(G))\) and the variational formulation of (AP) has a solution, then (AP) has a unique solution. The map \(H^ 1_ 0(H)\times L^ r(S,W^{1,p}(G))\ni \{u_ 0,f\}\mapsto u\in W^{1,r}(S,H^ 1_ 0(G))\) is locally Lipschitzian for all r. Some pointwise estimates on solutions are also given.