Global Existence And Blow-up Of Pseudo-parabolic Equation Solutions

Pseudo-parabolic equations play a very important role in natural science,finance,physics and other aspects.It has indispensable research value and significance.In this paper,we mainly study Cauchy problem of two kinds of pseudoparabolic equations,in terms of their critical Fujita exponent,second critical exponent,the life span for blow up solutions and the blow up time of the solution are discussed.Using the critical Fujita exponent to distinguish global solution from the non-global solution index classification.The second critical exponent refers to the critical decay order used to describe the initial values of the global solution and non-global solution in the coexistence region.The life span of the solution is to estimate the blow up time of the non-global solution of the equation.First,we considers the Cauchy problem of pseudo-parabolic equation with inhomogeneous terms ut=Δu+kΔut+w(x)up(x,t).In[14],Li et al gave the critical Fujita exponent,second critical exponent and the life span for blow up solutions under w(x)=|x|σ with σ>0.We further generalize the weight function w(x)～|x|σ for-2<σ<0,and discuss the conditions for the global and nonglobal solutions to obtain the critical Fujita exponent pc=1+(2+σ)/n.By constructing appropriate test function and constructing the lower solution of the equation,the equation using the comparison principle has a unique non-negative solution of blow up phenomenon.The global existence of the solution of the equation is obtained by using the compression mapping principle.Secondly,the Cauchy problem of the pseudo-parabolic equation ut=Δu+kΔut+‖w(x)u‖Lq(Rn)p-1 ur+1(x,t)with nonlocal source is studied,and the asymptotic behavior of its solution is discussed,where kernel function w(x)～|x|-s(x is large enough).Recently,Yang[431 et al.obtained the critical Fujita exponent Pc=1+(2q-nqr)/(nq-(n-qs)+).This paper continue studies the second critical exponent,obtain a*=(2q+(p-1)(n-qs)+)/(q(p+r-1))under two cases of w(x)∈ Lq and w(x)? Lq,i.e.the critical decay order of the initial data to identify the global solutions and non-global solutions in the coexistence area p>pc.Furthermore,the life span Tλ～λ-(2q(p+r-1))/((n-qs)+(p-1)+2q-q(p+r-1)min{a,n}),λ→0 of the blow-up solution is estimated.It is revealed that the third order viscous term kΔut and the weight function w(x)affect the asymptotic behavior of the solution,which is beneficial to the global existence of solutions or to delay the blow up time of the solution.