LOCAL SOLVABILITY AND DESTRUCTION OF SOLUTIONS TO NONLINEAR CAUCHY PROBLE AND INITIAL- BOUNDARY VALUE PROBLEMS from the Theory of Waves in Plasma

Three nonlinear Cauchy problems will be considered for the equations of ion-acoustic and drift waves in a plasma, as well as an initial boundary value problem for spherically symmetric ion-acoustic waves in a plasma. All equations in these problems are united by a common linear part. At first the fundamental solution of this linear part is constructed, and then the second and third Green's formulas are derived. Then the volume and surface potentials arising in Cauchy problems are considered and their properties are studied. For volume potential and potentials with weight a priori estimates of Schauder type are proved. Next, the Cauchy problems and the initial-boundary value problem are reduced to equivalent integral equations. For all problems, local time solvability is proved using the contraction mapping method. For two Cauchy problems, the existence of non-extendable solutions is proved, and for the third, the existence of a local solution in time is proved. For one from Cauchy problems using the modified method of H.A. Levin obtained sufficient conditions for the destruction of a solution in a finite time and found an upper estimate for the time of destruction of the solution. For another Cauchy problem using the nonlinear capacity method of Pokhozhaev obtained a result on the destruction of a solution in finite time and two results on the absence of even local solutions, and also obtained an upper estimate for the time of destruction of a solution. Finally, for the initial-boundary value problem, using the test function method, a result on the blow-up of the solution was obtained and an upper estimate for the blow-up time was obtained.