## Construction of a blow-up solution for a perturbed nonlinear heat equation with a gradient and a non-local term.(English)Zbl 1454.35221

Summary: We consider in this paper a perturbation of the standard semilinear heat equation by a term involving the space derivative and a non-local term. We prove the existence of a blow-up solution, and give its blow-up profile. Our proof relies on the following method: we linearize the equation (in similarity variables) around the expected profile, then, we control the nonpositive directions of the spectrum thanks to the decreasing properties of the kernel. Finally, we use a topological argument to control the positive directions of the spectrum.

### MSC:

 35K58 Semilinear parabolic equations 35R09 Integro-partial differential equations 35B44 Blow-up in context of PDEs 35B20 Perturbations in context of PDEs
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### References:

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