Critical exponent of non-global solutions for an inhomogeneous pseudo-parabolic equation with space-time forcing term

We investigate the critical exponent of non-global solutions to the following inhomogeneous pseudo-parabolic equation with a space-time forcing term: {u(t) - k Delta u(t) = Delta u + vertical bar u vertical bar(p) + t(sigma)omega(x) for x is an element of R-n, t > 0, u(x, 0) = u(0)(x) for x is an element of R-n, where n >= 1 is an integer; k > 0, p > 1, and sigma > -1 are three constants; and u(0), omega is an element of C-0(R-n). By obtaining a priori estimate for the solutions and the contradiction argument, we show that there exists a critical exponent: p(c)(sigma) :={2 sigma-1/2 sigma+1, if n = 1 and sigma is an element of(-1,- 1/2), infinity, if n = 1 and sigma is an element of(-1/2, infinity), 1-1/sigma, if n = 2 and sigma is an element of(-1, 0], infinity, if n = 2 and sigma is an element of[0, infinity), 2 sigma-n/2 sigma-n+2, if n > 2 and sigma is an element of(-1, 0], infinity, if n > 2 and sigma is an element of(0, infinity), such that the problem does not admit any global solutions when p < p(c)(sigma) and integral(Rn)omega(x)dx > 0. Our obtained results show that the forcing term induces an interesting phenomenon of continuity/discontinuity of the critical exponent p(c)(sigma) depending on the dimension n. Namely, we found that when n = 1, lim(sigma ->-1/2-)p(c)(sigma) = lim(sigma ->-1/2+)p(c)(sigma) = infinity; when n = 2 lim(sigma -> 0-) p(c)(sigma) = lim(sigma -> 0+) p(c)(sigma) = infinity; and when n >= 3 lim(sigma -> 0-) p(c)(sigma) = n/n-2 < infinity, lim(sigma -> 0+) p(c)(sigma) = infinity. Furthermore, lim(sigma ->kappa-) p(c)(sigma) with kappa = -1/2 when n = 1 and kappa = 0 when n >= 2 coincides with the critical exponent of the above problem with sigma = 0.