Abstract:
We consider two model non-linear equations describing electric oscillations in systems with distributed
parameters on the basis of diodes with non-linear characteristics. We obtain equivalent integral equations for
classical solutions of the Cauchy problem and the first and second initial-boundary value problems
for the original equations in the
half-space $x>0$. Using the contraction mapping principle, we prove the local-in-time
solubility of these problems.
For one of these equations, we use the Pokhozhaev method of non-linear capacity
to deduce a priori bounds giving rise to finite-time blow-up results and obtain upper bounds for the blow-up
time. For the other, we use a modification of Levine's method to obtain sufficient conditions for blow-up
in the case of sufficiently large initial data and give a lower bound for the order of growth of a functional
with the meaning of energy. We also obtain an upper bound for the blow-up time.
Keywords:non-linear equations of Sobolev type, destruction, blow-up, local solubility, non-linear capacity,
bounds for the blow-up time.
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