In this paper we consider the initial boundary value problem of a higher order viscous diffusion equation with gradient dependent potentials Φ（s） and sources A（s）. We first show the general existence and uniquene...In this paper we consider the initial boundary value problem of a higher order viscous diffusion equation with gradient dependent potentials Φ（s） and sources A（s）. We first show the general existence and uniqueness of global classical solutions provided that the first order derivatives of both Φ（s） and A（s） are bounded below. Such a restriction is almost necessary, namely, if one of the derivatives is unbounded from below, then the solution might blow up in a finite time. A more interesting phenomenon is also revealed for potentials or sources being unbounded from below. In fact, if either the source or the potential is dominant, then the solution will blow up definitely in a finite time. Moreover, the viscous coefficient might postpone the blow-up time. Exactly speaking, for any T 〉 0, the solution will never blow up during the period 0 〈 t 〈 T, so long as thc viscous coefficient is large enough.展开更多
基金Supported by the NSF of China(Grant Nos.11371153,11471127,11571062,11671155 and 11771156)NSF of Guangdong(Grant No.2016A030313418)+1 种基金NSF of Guangzhou(Grant Nos.201607010207 and 201707010136)the Fundamental Research Funds for the Central Universities(Grant No.DUT16LK01)
文摘In this paper we consider the initial boundary value problem of a higher order viscous diffusion equation with gradient dependent potentials Φ（s） and sources A（s）. We first show the general existence and uniqueness of global classical solutions provided that the first order derivatives of both Φ（s） and A（s） are bounded below. Such a restriction is almost necessary, namely, if one of the derivatives is unbounded from below, then the solution might blow up in a finite time. A more interesting phenomenon is also revealed for potentials or sources being unbounded from below. In fact, if either the source or the potential is dominant, then the solution will blow up definitely in a finite time. Moreover, the viscous coefficient might postpone the blow-up time. Exactly speaking, for any T 〉 0, the solution will never blow up during the period 0 〈 t 〈 T, so long as thc viscous coefficient is large enough.
