Generalized solutions of coagulation equations (Q1814367)

The author considers Smoluchowski's kinetic equation, modelling coagulation of particles of a dispersion system with process control of condensation in absence of exterior sources of particles \[ f_ t(m,t)=(r(m,t)f(m,t))_ m=2^{-1}\int^ m_ 0\varphi(m-m_ 1,m_ 1)f(m-m_ 1,t)f(m_ 1,t) dm_ 1-f(m,t) \] \[ \int_ 0^{+\infty}\varphi(m,m_ 1)f(m_ 1,t)dm_ 1+q(m,t),\;m,t\in R^ +_ 1. \tag{1} \] In (1), \(f(m,t)\geq 0\) is the distribution function of the particles at the moment \(t\in R^ +_ 1\) with masses \(m\in R^ +_ 1\); \(r(m,t)\geq 0\) is the velocity of growth of particles in condensation time; \(q(m,T)\geq 0\) is the spectrum of the particles entering in the system on the unit time from an exterior source. The given function \(\varphi(m,m_ 1)\geq 0\) is said to be the kernel of the coagulation; it characterizes the intensity of the coagulation of the particles of masses \(m\) and \(m_ 1\) in the unit time. Suppose (2) \(f(m,0)=f_{01}(m)\geq 0\), \(f(0,t)=f_{02}\geq 0\), \(m,t\in R^ +_ 1\). The author enounces two results concerning the existence of generalized solutions of the problem (1), (2). One of the results is the following. A family \((\mu_ t)_{t\geq 0}\) of positive Borel measures is said to be a generalized solution of the problem (1), (2) if all \(\mu_ t\), \(t\geq 0\), satisfy a certain condition given in the paper. If \(r(m,t)\geq 0\) is a locally Lebesgue integrable function, \(r(m,t)\leq k(1+m)\) in \(R^ +_ 2\), \(k=\)constant; \(f_{02}\in L^ +_ 1([0,T])\cap L^ +_ \infty([0,T])\); \(\mu_ 0\) and \(\nu_ t\) are positive Borel measures with bounded zero and first moments on \(R^ +_ 1\), \(t\in[0,T]\); the symmetric function \(\varphi\geq0\) is locally Lebesgue integrable on \(R^ +_ 2\) and satisfies \(\varlimsup_ m\varphi(m,m_ 1)m^{-\alpha}=H(m_ 1)\), \(0<\alpha<1\), \(m_ 1\in R^ +_ 1\) where \(H\) is a locally bounded function, then there exists a generalized solution of the problem (1), (2).