Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  consider a pseudo-parabolic equation with nonlinearities of variable exponent type                u    t    −  ν  △    u    t    −  div   (    |  ∇  u  |    m   (  x  )   −  2    ∇  u  )   =    |  u  |    p   (  x  )   −  2    u  ,      in    Ω  ×   (  0  ,  T  )   ,           associated with initial and Dirichlet boundary conditions. By means of a differential inequality technique,  obtain an upper bound for blow-up time if variable exponents   p(⋅)     p   (  ⋅  )        ,   m(⋅)     m   (  ⋅  )         and the initial data satisfy some conditions. Also, a lor bound for blow-up time is determined under some other conditions.
['Huafei Di', 'Yadong Shang', 'Xiaoming Peng']
I am writing a research paper on a idea Blow-up phenomena for a pseudo-parabolic equation with variable exponents
[{"from": "human", "value": "I am writing a research paper on a idea Blow-up phenomena for a pseudo-parabolic equation with variable exponents"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  consider a pseudo-parabolic equation with nonlinearities of variable exponent type                u    t    \u2212  \u03bd  \u25b3    u    t    \u2212  div   (    |  \u2207  u  |    m   (  x  )   \u2212  2    \u2207  u  )   =    |  u  |    p   (  x  )   \u2212  2    u  ,      in    \u03a9  \u00d7   (  0  ,  T  )   ,           associated with initial and Dirichlet boundary conditions. By means of a differential inequality technique,  obtain an upper bound for blow-up time if variable exponents   p(\u22c5)     p   (  \u22c5  )        ,   m(\u22c5)     m   (  \u22c5  )         and the initial data satisfy some conditions. Also, a lor bound for blow-up time is determined under some other conditions."}]