Critical behavior of a semilinear time fractional diffusion equation with forcing term depending on time and space
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Abstract
0, while the global solution exists for suitably small initial data u0 and w belonging to certain Lebesgue spaces when p≥1+2ααN−2σ−2α. (ii) The solution of the above superdiffusion equation blows up in finite time when 1
0, while the global solution exists for suitably small initial data u0 and w belonging to certain Lebesgue spaces when p>1+2ααN−2σ−2α and u1=0. (iii) For α+σ≥1, the solution of the above superdiffusion equation blows up in finite time when 1
0, while the global solution exists for suitably small initial data u0, u1 and w belonging to certain Lebesgue spaces when p>1+2ααN−2σ−2α. (iv) For α+σ<1, the solution of the above superdiffusion equation blows up in finite time when 1
0, while the global solution exists for suitably small initial data u0, u1 and w belonging to certain Lebesgue spaces when p>1+2ααN−2. The critical exponent in (iv) is different from that in (iii) and (ii). This peculiarity is related to the fact the time order of the equation and the inhomogeneous are both fractional, and so the role played by the second data u1 becomes “unnatural” as α∈(1,2). Namely, the change of the critical exponent in (iv) is due to that α+σ<1 and u1⁄≡0.
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DOI: 10.1016/j.chaos.2023.114309
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Keywords
Global solution; Finite time blow-up; Critical exponent; Time fractional-diffusion equation; Forcing term depending on time and space;All these keywords.
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